Negative remainder mismatch: state the quotient rule

Keep both reports beside their rules

Negative remainder mismatch can come from two different quotient conventions even when both calculations preserve the division identity. In this exact paper example, the integer being divided is minus seven and the divisor is positive three. One report gives quotient minus three and remainder two. Another gives quotient minus two and remainder minus one. Before changing a sign, check what each report was instructed to produce. The example concerns stipulated mathematical rules, without a claim about any calculator, spreadsheet or programming language.

Write the identity as a = bq + r, where a is the integer being divided, b is the positive integer divisor, q is an integer quotient and r is an integer remainder. The letters name their roles. Here a is minus seven and b is three. A report must preserve those original inputs while its quotient and remainder are inspected. Replacing minus seven with seven would test another problem and would hide the sign question that brought the two reports into conflict.

Check the arithmetic before diagnosing the convention

For the first report, three times minus three is minus nine. Adding the proposed remainder two gives minus seven. Its complete identity is therefore minus seven equals three times minus three plus two. For the second report, three times minus two is minus six. Adding minus one also gives minus seven. Its identity is minus seven equals three times minus two plus minus one. Both reports pass the stated identity check with the same original integer and divisor.

That agreement does not yet decide which report answers a particular request. The identity by itself permits other integer pairs as well. Quotient zero with remainder minus seven also gives minus seven, for example, but neither convention considered here would accept it. To specify a quotient-and-remainder answer, add the rule that selects an allowed pair. Treat the identity and that selection rule as separate checks, and retain both alongside the reported numbers.

State a nonnegative remainder rule

The first convention requires 0 <= r < b. With positive divisor three, the permitted integer remainders are zero, one and two. The first report has remainder two, so it meets this range as well as the identity. The second has remainder minus one and fails this particular range. Under a request for a nonnegative remainder smaller than three, the accepted pair is therefore quotient minus three and remainder two. The word smaller supplies a strict upper bound: remainder three is not permitted.

You can recover that quotient directly from the remainder. Subtract two from minus seven to obtain minus nine, then divide by three to obtain minus three. This check uses ordinary division by a nonzero number. It does not redefine a negative number as a positive one or discard a sign. The remainder is positive because the chosen quotient contributes minus nine, which is two below the original minus seven. Adding two restores the original integer exactly.

State a quotient toward zero rule separately

The second convention instead chooses q by truncating the exact fraction a divided by b toward zero, then defines r as a minus bq. Minus seven divided by three lies between minus three and minus two. Its toward-zero integer is minus two. With that quotient, r equals minus seven minus minus six, which is minus one. This produces the second report. The negative remainder is a consequence of this stated quotient rule, rather than an arithmetic failure in the identity.

Here toward zero means removing the fractional part without moving farther from zero. It is not the same instruction as choosing the greatest integer no larger than the exact fraction. That latter integer is minus three for this example, whereas truncation gives minus two. Positive fractions can make the two instructions look interchangeable because they agree there. Keep the precise instruction beside a negative-input test instead of deciding the rule from a few positive examples.

Convert the pair together when the required rule changes

Suppose the retained report is quotient minus two and remainder minus one, but the requested result needs the nonnegative range. Decrease its quotient by one, obtaining minus three. Increase its remainder by the divisor three, obtaining two. These two changes preserve the identity together: reducing the quotient contribution by three is offset by adding three to the remainder. The corrected report for that request is the pair minus three, two, with the original inputs still minus seven and positive three.

Changing only the remainder from minus one to two would leave quotient minus two unchanged. Three times minus two plus two equals minus four, so that isolated edit breaks the identity. Likewise, decreasing only the quotient gives three times minus three minus one, equal to minus ten. Keep both components in a correction note. A report is a quotient-and-remainder pair, and an adjustment that makes one field look acceptable can make the complete decomposition false.

The same adjustment has a general boundary for these two conventions. When a is negative and not an exact multiple of positive b, the toward-zero remainder is negative and has magnitude less than b. Decreasing its quotient by one and adding b to its remainder puts the new remainder strictly between zero and b. The identity remains unchanged. This statement is restricted to positive divisors and the two explicitly defined rules; it is not an instruction to add the divisor to every negative-looking number in an unfamiliar record.

Use controls that agree and one that separates the rules

For positive seven divided by three, both conventions give quotient two and remainder one. The exact fraction lies between two and three, so its toward-zero integer is two, and remainder one meets the nonnegative range. This passing control is useful for checking basic arithmetic, but it cannot establish that the two rules will agree on negative inputs. Their difference is absent from this particular input, so a test limited to seven misses the mismatch we are diagnosing.

For minus six divided by positive three, both conventions give quotient minus two and remainder zero. An exact multiple has no fractional quotient part to remove and no negative nonzero remainder to adjust. Zero divided by three gives quotient zero and remainder zero under both rules too. Together with the minus-seven case, these controls identify the relevant boundary: negative input with a nonzero division remainder separates the two conventions here. Preserve those input conditions when describing what the tests establish.

Explain why the nonnegative answer is unique

Assume two integer pairs both preserve a = bq + r and both meet 0 <= r < b for the same positive divisor. Subtracting their identities makes b times the difference of the quotients equal to the difference of the remainders, up to the corresponding sign. Each remainder lies within the same interval of width b, so their difference has absolute value strictly less than b. The only integer multiple of positive b with that property is zero.

The quotient difference must consequently be zero, and then the remainder difference must be zero as well. Thus the two pairs are identical. This argument explains why applying both the identity and the nonnegative range settles the answer uniquely. It does not say that the identity alone identifies a pair, as our two minus-seven reports already show. Nor does it extend the selected range unchanged to negative divisors, which are outside this worksheet.

Keep the correction tied to the requested output

An adequate diagnosis can read: the report uses a toward-zero quotient, while the request requires a nonnegative remainder smaller than the positive divisor; adjust quotient and remainder together, then recheck both conditions. If the request instead specifies the toward-zero convention, the second report already answers it. There is no reason to overwrite that report merely because another exercise uses a different definition. Where the rule is absent, record that missing instruction before selecting one convention on the reader's behalf.

Bound the optional tabletop accounts

The Dancing Branches placemat account shows four rounded-corner rectangular mats with multicolored leaves and dark branches. The image does not authenticate the stated maker or model, dimensions, material or performance. Its printed design supplies no arithmetic input.

The Emerging placemat account shows four fanned rectangular mats with pink grounds and flowering branches. Maker, cork backing, coating, dimensions and heat or stain resistance are not independently verified. No cleaning or protection-performance claim is adopted.

These accounts provide optional tabletop context only. Neither placemat supplies a quotient rule or the paper numbers, and no purchase is needed to inspect either report.

Keep neighboring remainder questions distinct

The negative base two notation guide chooses parity digits and divides by minus two to construct a canonical string. Here the divisor is positive three and the task is to diagnose quotient conventions. There is no binary encoding or digit-construction procedure.

The greatest common divisor on paper guide preserves common divisors through repeated remainder steps and relates the result to a regular square tiling. This worksheet does not compute a greatest common divisor or use a tiling model.

Keep payment instructions with their transaction

If you separately consider a placemat order on republeq.com, assess the current offer and transaction before choosing how to pay. The signed integers in this paper exercise are defined mathematical inputs, not product quantities, prices or cryptocurrency amounts. Before using USDT, inspect the actual checkout and determine what that particular order presents. Check the requested amount and denomination, the receiving destination displayed, any network instruction, and the order status before sending funds. These are questions to resolve, rather than statements that an option exists for either offer. Confirm that the item and intended quantity match your proposed order. Resolve product and supplied-content questions separately from payment instructions. A photograph does not authenticate transaction terms. Keep the current instructions and the result of any action you actually take, distinguishing a pending step from a completed payment. If the selection changes while you are checking, review the resulting transaction instead of carrying an earlier conclusion across automatically. Determine what the merchant currently supplies about delivery, cancellation and any refund process before relying on it. Do not assume a transfer can be reversed or a particular network will be supported. This conditional checklist makes no claim about token acceptance, transfer success or available remedies. You can complete the paper worksheet without buying either offer. A correct remainder does not authenticate a payment destination or show that an order has been accepted. Never reinterpret an actual amount or order reference using this worksheet's adjustment. The purchase needs its own current evidence, and its records belong separately from the invented arithmetic exercise.

Retain the identity and the selection rule

For minus seven divided by positive three, preserve both valid identities before recording which rule the requested output uses. The nonnegative-remainder convention selects minus three, two; the toward-zero quotient convention selects minus two, minus one. Check any conversion by adjusting the pair together and testing the original identity and required range again. Keep positive seven, exact multiple minus six and zero as agreement controls. This record explains the mismatch without inventing a device fault or treating one convention as an arithmetic error under every possible instruction.