Remainder Calculator

Divide integers exactly and compare nonnegative Euclidean remainders with truncation-toward-zero remainders.

Calculation inputs

Precision & interpretation

Both conventions are shown explicitly. Zero is not an allowed divisor. The grouping sketch uses a bounded number of representative blocks rather than drawing billions of groups.

Calculation results

Remainder r2Euclidean remainder · 0 ≤ r < |b|

Integer quotient q3

Reconstruction17 = 5 × (3) + (2)

Other conventionq = 3, r = 2

Step-by-step solution

Choose the formula
$a = bq + r$
Substitute your inputs
$17 = 5q + r$
Calculate in order
$q = 3; r = 17 − (5) × (3)$
Write the result
$17 = 5 × (3) + (2)$

Division identity

Dividend17Divisor5Quotient3Remainder2

Identity: a = bq + r. Blocks represent quantities; their on-screen sizes do not encode magnitude.

Exact integer reconstruction. The Euclidean remainder is nonnegative for either divisor sign.

Exact value & calculation details

Quantity	Value
Dividend	17
Divisor	5
Quotient	3
Remainder	2

Understand the calculation

Concept & meaning

Integer division writes a dividend as a whole-number multiple of the divisor plus a remainder. The reconstruction a = bq + r is checked with exact integer arithmetic.

Formula & special cases

In Euclidean division the remainder is nonnegative and smaller than |b|. With a negative dividend, the quotient may differ from a programming-language quotient truncated toward zero.

Precision & limitations

Both conventions are shown explicitly. Zero is not an allowed divisor. The grouping sketch uses a bounded number of representative blocks rather than drawing billions of groups.

a = b q + r; 0 \leq r < | b |

Worked example from your inputs

Euclidean remainder · 0 ≤ r < |b| = 2

Frequently asked questions

Why can a negative dividend give a positive remainder?

Euclidean division chooses the quotient so that 0 ≤ r < |b|. For example, −17 = 5 × (−4) + 3.

How does truncation differ?

It rounds the quotient toward zero; its nonzero remainder has the same sign as the dividend. For example, −17 = 5 × (−3) − 2.

Can the divisor be negative?

Yes. Euclidean remainders still satisfy 0 ≤ r < |b|; the quotient sign follows from a = bq + r.

Why is division by zero rejected?

No unique quotient-and-remainder decomposition exists with a zero divisor.

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What does the Remainder calculate?

Divide integers exactly and compare nonnegative Euclidean remainders with truncation-toward-zero remainders.

Which values should I enter?

Use the labeled fields above. Whole integer, absolute value up to 10¹². Values outside the supported range clear the result and show an explanation.

What formula does this calculator use?

a = bq + r; Euclidean: 0 ≤ r < |b|

Why might my result differ from another source?

Results can differ when another source uses different units, rounding rules, conventions or assumptions. Compare the inputs and definitions before comparing results.

Does the result update automatically?

Yes. Inputs and presets update the result, teaching diagram and expandable steps together. Invalid input clears the previous result.

Sources & References

$NIST official logo$ NIST DLMF · CongruencesInteger divisibility and number-theory notationRead the reference $NIST official logo$ NIST DLMF · Number theoryPrime powers and divisor functionsRead the reference

References explain the mathematics; they do not imply institutional endorsement.