Divisibility Test

What question does a divisibility test answer?

The calculator starts with two roles that are easy to reverse. The dividend is the number being divided. The divisor is the number that describes the group size or the proposed factor. In the expression 84 divided by 7, 84 is the dividend and 7 is the divisor. The central question is whether there is an integer quotient q such that dividend equals divisor times q. If such a q exists, the divisor divides the dividend and the calculator calls the pair Divisible. If no integer q exists, the pair is Not divisible.

Divisibility is an exact yes-or-no property, not a judgment about whether a decimal answer looks close to a whole number. For example, 84 divided by 7 is exactly 12, so 7 divides 84. By contrast, 85 divided by 7 is 12 and a little more, so 7 does not divide 85. The calculator makes that difference visible by reporting the remainder as well as the status. A zero remainder is the decisive condition; a nonzero remainder is a compact witness that equal whole-number groups do not use up the dividend exactly.

The word whole number in this page means an integer with no fractional part. The dividend may be positive, zero, or negative. The divisor has a narrower contract: it must be a positive whole number. That choice keeps the question unambiguous and avoids division by zero. The result is arithmetic, so it does not know whether a number represents books, points, pixels, people, or something else. You supply the meaning, while the calculator checks the integer relationship.

• Dividend: the signed whole number being tested.
• Divisor: the positive whole-number candidate that may divide the dividend.
• Remainder: what is left after complete divisor-sized groups are accounted for.
• Divisibility: Divisible exactly when the reported remainder is zero.

Enter the two inputs within the calculator contract

The signed whole-number dividend field accepts a safe value from -1,000,000,000,000 through 1,000,000,000,000. A leading minus sign is allowed, but a decimal such as 12.5 is not a whole number and does not belong in this calculator. The positive whole-number divisor field accepts a safe value from 1 through 1,000,000. It cannot be zero or negative. Both values are treated as numbers rather than measurements with automatic unit conversion, so the calculation is meaningful only when the two inputs describe the same counting scale.

The word safe matters in a programming implementation. Ordinary JavaScript numbers can represent many integers exactly, but not every arbitrarily large integer. This page keeps the dividend well inside the supported safe range and rejects values that are not finite, integral, and safe. A value written in scientific notation may be interpreted by an input control as a number, but the mathematical value still has to be a whole number inside the stated range. Do not use a rounded large value as a substitute for an exact integer that lies outside the contract.

A positive divisor can be larger than the absolute value of a nonzero dividend. That is valid, although it usually produces a nonzero remainder and therefore a Not divisible result. For instance, 4 tested with divisor 9 is a legitimate request. The divisor is not required to be smaller than the dividend, and the dividend is not required to be positive. The only prohibited divisor value is zero, because no quotient and remainder relationship can use zero as the divisor in this test.

• Valid examples include dividend -84 with divisor 7, dividend 0 with divisor 13, and dividend 29 with divisor 1.
• Invalid examples include dividend 12.5, divisor 0, divisor -4, and a value outside either field's numeric bounds.
• Enter counts as whole numbers; decimals, rounding assumptions, and hidden unit conversions are outside this page's contract.

The formula: divide, then inspect the remainder

The catalog formula is dividend mod divisor = 0 for a divisible pair. In more complete division notation, let a be the dividend, d be the positive divisor, q be the integer quotient, and r be the remainder. They are related by a = d x q + r. The calculator evaluates a mod d to obtain r, then checks whether r equals zero. It does not need to search for all possible factors or construct a factorization; one remainder calculation answers this particular question.

When the dividend is positive, the familiar picture is straightforward. Dividing 100 by 6 gives q = 16 and r = 4 because 100 = 6 x 16 + 4. Since 4 is not zero, 6 does not divide 100. Dividing 84 by 7 gives q = 12 and r = 0 because 84 = 7 x 12 + 0. The zero in that identity is precisely why the status changes to Divisible.

The quotient is not a separate input and is not one of the displayed result fields. The result panel shows Remainder and Divisibility, while the calculation step shows the modulo expression. If you need the quotient for a written solution, reconstruct it from q = (a - r) / d. The result is an integer because r came from the calculator's remainder operation. This formula is also a useful independent check: multiply the divisor by the inferred quotient and add the displayed remainder to recover the original dividend.

The remainder convention becomes important for negative dividends. In this calculator, a nonzero remainder keeps the dividend's sign, so the identity still holds with a negative r. That is a deliberate part of the catalog explanation. The zero test is unaffected by whether a programming language chooses a signed or nonnegative representative, but the displayed remainder and the corresponding quotient can differ.

• Division identity: dividend = divisor x quotient + remainder.
• Divisible condition: remainder = 0.
• Not divisible condition: remainder is any nonzero integer allowed by the signed convention.
• Inferred quotient: (dividend - remainder) / divisor.

How to interpret quotient and remainder

A quotient tells you how many complete divisor-sized groups fit under the calculator's division convention. A remainder tells you the signed amount left after those groups. For 100 and 6, the quotient is 16 and the remainder is 4. The statement does not mean that 100 is made of only 16 groups; it means 16 complete groups account for 96 and 4 remains. The ordinary fractional quotient is 16 and 4/6, or 16 and 2/3, but the calculator intentionally reports the integer remainder instead of a decimal approximation.

For 85 and 7, the inferred quotient is 12 and the remainder is 1. Rebuilding the dividend gives 7 x 12 + 1 = 85. That reconstruction is often clearer than looking at a decimal such as 12.142857. The calculator's status says Not divisible because the leftover amount is not zero. A nonzero remainder does not mean that division failed; every allowed pair has a quotient and remainder. It means exact whole-number grouping did not finish with nothing left.

For positive inputs, the quotient is the usual whole-number part and the remainder is nonnegative and smaller than the divisor. For a negative dividend, use the displayed signed remainder when reconstructing the quotient. With -17 and 5, the remainder is -2, so q = (-17 - (-2)) / 5 = -3. The identity is -17 = 5 x (-3) + (-2). This is the quotient associated with the calculator's signed remainder output, not a request to round a decimal quotient toward positive or negative infinity by guesswork.

The calculator reports the remainder rather than the quotient because the remainder is the shortest direct proof of divisibility. If the status is Divisible, the remainder is zero and the quotient can be recovered exactly. If the status is Not divisible, the nonzero remainder tells you how far the division is from an exact grouping under the selected signed convention.

• A zero remainder means the quotient is an integer and no amount is left over.
• A nonzero remainder means complete groups exist, but they do not account for the entire dividend.
• Always use the displayed remainder in q = (a - r) / d, especially when a is negative.
• Do not confuse a remainder with the decimal fraction after the decimal point; they are related but formatted differently.

Worked examples with positive dividends

Start with the default-style example 84 and 7. Seven groups of 12 account for 84, so the division identity is 84 = 7 x 12 + 0. The remainder is 0 and the status is Divisible. This is the ideal case for checking an equal split: 84 objects can be placed into groups of 7 with no object left over, or into 12 groups with 7 objects per group. Which story you tell depends on whether the divisor represents group size or number of groups, but the arithmetic relationship is the same.

Now change only the dividend to 85. The largest complete multiple of 7 below 85 is 84, which is 7 x 12. One remains, so 85 = 7 x 12 + 1. The calculator returns remainder 1 and Not divisible. Notice that the quotient did not change when one was added to the dividend, but the remainder did. That small change is enough to change the yes-or-no answer.

For a less familiar divisor, test 221 with 13. Since 13 x 17 = 221, the remainder is 0 and the result is Divisible. This illustrates why the exact calculator is useful when no quick classroom rule comes to mind. With 222 and 13, the same quotient of 17 accounts for 221 and the remainder is 1, so the status becomes Not divisible. The pair 221 and 13 also reminds us that a divisor need not be 2, 3, 5, or 10 to be tested exactly.

Consider 1000 with divisor 16. Sixteen x 62 is 992, leaving 8. The result is Not divisible, even though 1000 is a round decimal-looking number. A round appearance is not a divisibility proof. Finally, 4 with divisor 9 returns remainder 4 and Not divisible: when a positive dividend is smaller than a larger divisor, zero complete groups fit and the dividend itself remains as the remainder.

• 84 mod 7 = 0: quotient 12, remainder 0, Divisible.
• 85 mod 7 = 1: quotient 12, remainder 1, Not divisible.
• 221 mod 13 = 0: quotient 17, remainder 0, Divisible.
• 222 mod 13 = 1: quotient 17, remainder 1, Not divisible.
• 1000 mod 16 = 8: quotient 62, remainder 8, Not divisible.
• 4 mod 9 = 4: quotient 0, remainder 4, Not divisible.

Negative dividends and signed remainders

A negative dividend is still a whole number, and divisibility applies to it just as it applies to a positive integer. For example, -84 is divisible by 7 because -84 = 7 x (-12) + 0. The calculator therefore returns remainder 0 and Divisible. The sign of the quotient changes, but zero remains zero, so an exact negative multiple is not a special exception.

For a negative dividend that is not an exact multiple, this calculator retains the sign of the dividend in the nonzero remainder. Test -17 with divisor 5. The handler's remainder is -2, and the associated identity is -17 = 5 x (-3) + (-2). Because -2 is nonzero, the status is Not divisible. Test 17 with divisor 5 and the remainder is 2 instead. Both pairs are nondivisible, but the displayed remainders have opposite signs because the dividends have opposite signs.

Another useful example is -18 with divisor 5. The signed remainder is -3 and the inferred quotient is -3: 5 x (-3) + (-3) = -18. For -4 with divisor 9, zero complete groups fit under truncation toward zero, so the remainder is -4 and the status is Not divisible. These results may look different from a convention that always chooses a remainder from 0 through d - 1. Both conventions can describe the same divisibility fact, but this page uses the signed result produced by its handler.

If you need a nonnegative representative for a separate modular-arithmetic exercise, a negative remainder can be shifted by adding the positive divisor. For -17 and 5, -2 and 3 represent the same residue class because they differ by 5. That shift is an interpretation step, not the value this calculator displays. The safe rule for this page is simple: read the returned sign as shown, and use only equality with zero to decide divisibility.

• -84 mod 7 = 0: the negative dividend is an exact multiple and is Divisible.
• -17 mod 5 = -2: the remainder is nonzero and signed, so the result is Not divisible.
• -18 mod 5 = -3: the identity is -18 = 5 x (-3) + (-3).
• -4 mod 9 = -4: a divisor larger than the absolute dividend leaves the signed dividend as the remainder.

Zero, one, and size-related edge cases

Zero is a valid dividend. Every positive divisor divides zero because zero equals that divisor multiplied by the integer quotient 0. Thus 0 with divisor 13 gives remainder 0 and Divisible, and 0 with divisor 1 does the same. This is different from dividing by zero: the dividend may be zero, but the divisor may not. The calculator rejects divisor 0 rather than trying to assign a remainder to an undefined operation.

One is the simplest positive divisor. Every allowed dividend is divisible by 1 because a = 1 x a + 0. A negative dividend does not change that fact: -37 with divisor 1 has quotient -37 and remainder 0. If you are checking a user-entered divisor and it is 1, a Divisible result is expected for any valid dividend, including zero.

When the divisor equals a nonzero dividend, the result is also Divisible. For 7 with divisor 7, the quotient is 1 and the remainder is 0. For -7 with divisor 7, the quotient is -1 and the remainder is 0. When the positive divisor is larger than the absolute value of a nonzero dividend, the result is Not divisible because no nonzero whole-number multiple of that divisor can fit without overshooting. For a positive example, 4 mod 9 is 4; for a negative example, -4 mod 9 is -4 under the signed convention.

The numeric boundaries are edge cases too. A dividend of exactly -1,000,000,000,000 or 1,000,000,000,000 is within the stated bound, and a divisor of exactly 1,000,000 is allowed. Values just beyond those limits are not accepted even if the remainder would be easy to calculate. This boundary protects the exact integer behavior of the number-based handler.

• Dividend 0 with any allowed divisor: remainder 0 and Divisible.
• Divisor 1 with any valid dividend: remainder 0 and Divisible.
• Divisor 0: invalid, because division by zero is not defined.
• Divisor greater than a nonzero dividend's absolute value: the result is normally Not divisible.
• Endpoint values are valid only when they remain inside the catalog's stated bounds.

Mental divisibility rules versus the exact test

Mental rules are shortcuts for selected divisors. They are valuable in arithmetic practice because they reduce a large-looking question to a few digits, but each rule has a precise scope. The calculator is the general check for every positive divisor in its range. It returns the actual remainder, while a mental rule usually gives only a yes-or-no conclusion. When the divisor is unusual, such as 7, 13, or 37, the exact remainder test avoids inventing a shortcut that has not been justified.

For divisibility by 2, inspect the last digit: an even last digit is enough. For 5, the last digit must be 0 or 5. For 10, the last digit must be 0. Divisibility by 4 depends on the last two digits, and divisibility by 8 depends on the last three digits. These rules work because the place values before those final digits are multiples of the selected divisor. A number such as 728 is divisible by 8 because 728 divided by 8 is 91, while 1234 is not divisible by 4 because its last two digits, 34, are not divisible by 4.

For 3 and 9, add the decimal digits. The sum of the digits of 462 is 12, so 462 is divisible by 3 but not by 9. For 6, apply both the rule for 2 and the rule for 3: 462 is even and its digit sum is divisible by 3, so it is divisible by 6. For 11, take the alternating sum of the digits and check whether the result is a multiple of 11. The sign of a whole number does not alter whether it is divisible by these positive divisors, so these digit rules can be applied to the absolute value of a negative dividend.

A shortcut can still be misapplied. The digit-sum rule for 9 does not test divisibility by 7, and a last-digit check for 5 does not test divisibility by 15 without an additional condition. Even a correct rule may be inconvenient when the number is copied incorrectly or contains a decimal. Use a mental rule to estimate or verify a result, then use the calculator when you need the explicit remainder, an arbitrary divisor, or a dependable check of signed input.

• 2: the last digit is even.
• 3 or 9: the digit sum is divisible by 3 or 9, respectively.
• 4 or 8: inspect the last two or three digits, respectively.
• 5 or 10: use the appropriate last-digit condition.
• 6: the number must satisfy both the 2 and 3 tests.
• 11: the alternating digit sum must be a multiple of 11.

Common mistakes and how to catch them

The most common error is reversing the roles of the two inputs. Saying that 7 divides 84 is true, but the calculator expression is dividend 84 and divisor 7. Testing dividend 7 with divisor 84 asks a different question and returns a different remainder. Before calculating, read the pair as a sentence: does the divisor divide the dividend? That sentence keeps the direction visible.

Another mistake is treating a rounded decimal quotient as proof. A display of 12.0 may have been rounded from a value that is not an integer, and a display of 12.14 clearly contains a fractional part. The exact remainder is safer than visual inspection. Similarly, a number that looks round because it ends in zero is not automatically divisible by every convenient divisor. 1000 is divisible by 10 but not by 16, as the remainder example shows.

Negative results cause a different kind of confusion. Some references and programming languages represent the remainder of a negative dividend as nonnegative, while this handler retains a negative sign for a nonzero result. Do not mark -17 mod 5 as an error merely because you expected 3; on this page the returned remainder is -2. The divisibility status is still easy to read because both -2 and 3 are nonzero. Reconstruct the original dividend with the page's displayed remainder if you need to check the quotient.

Finally, do not treat a rejected input as a surprising arithmetic answer. A divisor of zero has no valid remainder, a decimal dividend is not a whole-number input, and an out-of-range value is outside the supported exact-number boundary. Correct the input rather than silently rounding, changing the sign, or substituting a nearby value.

• Check the direction: the divisor is tested against the dividend, not the other way around.
• Check zero, not a rounded decimal quotient, to decide divisibility.
• Keep the negative sign on a nonzero remainder when using this calculator.
• Do not turn a decimal, zero divisor, or out-of-range value into a nearby allowed value without saying so.
• Remember that Not divisible still means a valid division exists; it has a nonzero remainder.

The modulo operation in programming

In code, divisibility is commonly implemented by calculating a remainder and comparing it with zero. For integer variables named a and d, the basic condition is a modulo d equals 0, often written as a% d === 0 in JavaScript. The comparison is more reliable than checking whether a divided by d looks like a whole number after floating-point formatting. A zero remainder remains the decisive signal even when a is negative, provided d is nonzero.

The exact remainder value needs more care than the zero test. JavaScript's remainder operator keeps the dividend's sign for a nonzero result, which is the behavior represented by this calculator. Other languages may choose a nonnegative remainder when the divisor is positive, or may document a different quotient rule. If a program needs to reproduce this page's displayed remainder, test its language's operator with negative cases instead of assuming all modulo operators agree. If it only needs the boolean divisibility result, equality with zero is usually portable for integer values and a nonzero divisor.

The handler also illustrates why input validation belongs beside arithmetic. It accepts finite safe integers only, limits the dividend to plus or minus 1e12, and limits the positive divisor to 1e6. A program that accepts arbitrary text, decimals, infinity, or integers beyond its exact numeric representation can produce a result that looks plausible while using a changed value. For very large integer data, a text-to-integer approach such as BigInt may be appropriate, but that is a different implementation contract from this number-based calculator and should be tested separately.

A useful programming postcondition is to verify the division identity after calculating. Let r be the returned remainder and q be (a - r) / d. Confirm that q is an integer and that d x q + r equals a. A small test matrix should include a positive multiple, a positive nonmultiple, a negative multiple, a negative nonmultiple, zero, divisor one, and a divisor larger than the absolute dividend. These cases catch sign, zero, and boundary mistakes that a single positive example cannot reveal.

Modulo is useful beyond a yes-or-no page: it can select every seventh item, detect even and odd positions, rotate through a repeating pattern, or check whether a batch closes evenly. Those uses still require a clear definition of what happens to negative indexes and whether a nonnegative residue is needed. Do not silently transfer the signed output from this page into an indexing operation that only accepts values from 0 through d - 1; normalize deliberately when that separate contract requires it.

• For integer inputs, test divisibility with a remainder comparison rather than a rounded decimal comparison.
• Test negative values because remainder signs differ across programming environments.
• Keep validation aligned with the supported finite, safe, whole-number bounds.
• Use the identity a = d x q + r as a postcondition for a custom implementation.
• Normalize a negative residue only when the next program operation explicitly requires a nonnegative range.

Classroom and practical uses

In a classroom, the calculator can separate three ideas that are often compressed into one answer: the dividend, the divisor, and the remainder. A student can predict whether 84 is divisible by 7, calculate the quotient by hand, and then use the reported remainder to check the work. Changing 84 to 85 creates a controlled comparison: the complete groups stay the same while one leftover unit changes the status. Asking students to explain the identity behind each result is more useful than memorizing the word Divisible alone.

The page also supports lessons about factors and multiples. If a divisor returns remainder zero, it is a factor of the dividend. Testing several candidate divisors can reveal factors, although this calculator does not list every factor automatically. Digit rules can be compared with the general modulo test: a student might use the rule for 9 on 1233, then use an arbitrary divisor such as 13 to see why a general operation is needed. Negative examples extend the lesson beyond counting objects and show that divisibility is an integer property, not only a positive-number trick.

Outside schoolwork, the same relationship appears whenever whole items are grouped. A coordinator can check whether a count makes complete teams of a chosen size. A stock list can show whether units fit into equal packs. A repeating layout can use a remainder to identify a position in a cycle. In each case, the arithmetic result is only as meaningful as the input meaning: items must be counted as whole units, the divisor must describe the intended group size, and the two numbers must refer to the same total.

For a practical decision, treat a nonzero remainder as information rather than as an automatic instruction. A remainder of 1 may mean one item needs another group, one item remains unassigned, or the group size should be reconsidered. The calculator does not know which action is appropriate and does not estimate waste, cost, staffing, capacity, or safety. It simply establishes whether the entered integer relationship closes exactly.

• Use paired examples such as 84 and 85 to show how one leftover changes the status.
• Ask learners to reconstruct the dividend from divisor, quotient, and remainder.
• Compare a known mental rule with an arbitrary divisor to distinguish shortcuts from the general test.
• For grouping tasks, define the unit and group size before entering numbers.
• Treat the remainder as evidence for a decision, not as a recommendation about what to do next.

Limitations: what this calculator does not decide

This is an integer remainder test, not a full number-theory workspace. It does not return the quotient as a displayed result, enumerate all factors, calculate a greatest common divisor, or decide whether the dividend is prime. A zero remainder can show that one selected divisor is a factor, but it says nothing by itself about the rest of the factorization. If you need those other questions, they require their own inputs and formulas rather than a reinterpretation of this page's status.

The calculator does not accept decimal quantities, fractions, measurements with uncertainty, or automatic rounding. A physical count may have been estimated, and a report may show a rounded total, but entering that rounded total produces a test of the rounded integer. It does not prove that the underlying measurement or source data was exact. Do not use a divisibility result to imply precision that was not present in the original information.

For negative dividends, the page does not normalize every remainder into the interval from 0 through divisor minus 1. Its explanation intentionally preserves the sign of a nonzero remainder. That is suitable for reading the handler's result and deciding whether it is zero. It may not be the desired representation for a separate task such as array indexing, a residue table, or a convention that requires nonnegative representatives. Convert only after identifying that separate requirement.

The supported range is finite and the arithmetic is deterministic, but the result is not a prediction or a guarantee about a real process. It cannot check whether a proposed grouping fits a room, complies with a policy, or is safe for a particular operation. It cannot resolve ambiguous data definitions, missing items, duplicate records, or inconsistent units. Use the result as a precise mathematical fact about the two entered integers and keep broader conclusions tied to the information those integers actually represent.

• Not included: displayed quotient, complete factorization, factor list, greatest common divisor, or primality classification.
• Not included: decimals, automatic rounding, uncertainty analysis, or unit conversion.
• Not included: nonnegative remainder normalization for negative dividends.
• Not included: predictions, capacity decisions, policy decisions, or safety approval.

A dependable way to check any result

A short routine makes the calculator useful for both quick checks and written solutions. First, identify the dividend as the number being tested and the divisor as the positive group size. Second, confirm that both are whole numbers, that the divisor is not zero, and that both values fit the stated bounds. Third, estimate the answer or apply a mental rule if one is relevant, but keep that estimate separate from the exact result. Fourth, calculate and read the remainder before reading the status. Fifth, if the quotient matters, compute q = (a - r) / d using the displayed signed remainder. Sixth, verify a = d x q + r.

Suppose the inputs are -17 and 5. The validation passes because -17 is a signed whole number and 5 is positive. The calculator reports -2, so the status is Not divisible. The inferred quotient is (-17 - (-2)) / 5 = -3. Rebuilding the input gives 5 x (-3) + (-2) = -17. This check respects the calculator's convention and avoids replacing its signed remainder with an unrequested nonnegative representative.

Suppose the inputs are 0 and 13. The validation passes, the remainder is 0, and the status is Divisible. The quotient is (0 - 0) / 13 = 0, and the identity is 13 x 0 + 0 = 0. Suppose the inputs are 4 and 9. The remainder is 4, the inferred quotient is 0, and 9 x 0 + 4 = 4. These two small cases show why zero and a larger divisor deserve explicit attention instead of being treated as input mistakes.

The final habit is to report the result with enough context to be useful. Say which dividend and divisor were tested, give the remainder, state Divisible or Not divisible, and include the quotient only if you have inferred it consistently. A complete sentence such as 85 is not divisible by 7 because its remainder is 1 is clearer than a bare Not divisible label. It preserves the calculation that another reader needs in order to check your conclusion.

• Identify the two roles before entering numbers.
• Validate whole-number type, sign, nonzero divisor, and bounds.
• Read the exact remainder and use zero as the status condition.
• Infer and verify the quotient only when it is needed.
• State the numbers, remainder, and conclusion together when sharing the result.

Questions visitors often have

Does zero divide every number? No. The phrase is reversed in that question. Zero cannot be the divisor because division by zero is undefined. What is true here is that every allowed positive divisor divides the zero dividend. For example, 13 divides 0 because 0 = 13 x 0. Keeping dividend and divisor in their proper roles resolves this frequently repeated confusion.

Can a negative number be divisible? Yes. Divisibility asks whether an integer multiple exists, and the multiplier may be negative. The pair -84 and 7 is divisible because -84 = 7 x (-12). A negative nonmultiple can still be divided, but its remainder is nonzero and signed on this page. The sign does not make a zero remainder become nonzero or vice versa.

Why is the remainder negative for some negative inputs? This handler follows the remainder behavior in which the sign of a nonzero remainder follows the dividend. Thus -17 mod 5 is -2 here. Another convention may represent the same residue as 3, but the divisibility decision is unchanged because neither value is zero. Use the page's displayed value when reconstructing its quotient.

What if the divisor is bigger than the dividend? That is a valid input when the dividend is nonzero. For a positive dividend smaller than the divisor, the quotient is 0 and the remainder is the dividend, so the result is Not divisible. Zero is the exception: zero remains divisible by every allowed positive divisor.

Can I enter a decimal or a rounded measurement? Not for this calculator. The contract is for safe whole numbers. If a real-world quantity was rounded before you entered it, the result describes the rounded integer only. Keep the source data and its uncertainty in view rather than treating an integer remainder as a statement about unmeasured precision.

• Zero is a valid dividend; zero is not a valid divisor.
• Negative dividends are allowed, but divisors must remain positive.
• A negative nonzero remainder is expected for some negative inputs.
• A larger divisor is allowed and usually leaves the nonzero dividend as the remainder.
• Decimals and values outside the supported bounds need a different calculation contract.