Modulo calculator
Find the non-negative remainder of an integer division, verify the quotient identity, and export the current calculation to a genuine Excel workbook.
Inputs
Use whole numbers. The divisor must be greater than zero.
Required integer. Commas are allowed only as standard thousands separators.
Required positive integer. Zero and negative divisors are rejected.
Live results
Results update as soon as both entries form a valid integer division.
Remainder (r)
3
2,026 divided by 7 leaves 3.
Integer quotient (q)
289
Largest multiple
2,023
Divisibility
No
The remainder is 3; 2,026 is not divisible by 7.
Calculation detail
| Step | Value | Meaning |
|---|---|---|
| Dividend (x) | 2,026 | The integer being divided |
| Divisor (y) | 7 | The positive modulus |
| Integer quotient (q) | 289 | The whole-number groups of 7 |
| Largest multiple (q × y) | 2,023 | The greatest multiple of 7 not above 2,026 |
| Remainder (r) | 3 | The amount left after removing complete groups |
| Reconstructed dividend | 2,026 | Confirms q × y + r equals x |
The table and Excel workbook use the same unrounded integer model as the result cards.
How to use the modulo calculator
What this calculator does
This calculator finds the standard non-negative remainder produced when one integer is divided by a positive integer. It also shows the integer quotient, the largest complete multiple of the divisor contained in the dividend, a divisibility answer, and the identity that reconstructs the original dividend. It is an exact integer-arithmetic tool: it does not accept decimal values, estimate probabilities, choose a programming-language convention for negative divisors, or solve a general congruence equation with unknown variables.
When to use it
Use modulo when you need to test whether a quantity divides evenly, wrap a repeating counter such as weekdays or clock positions, distribute objects into equal groups and identify leftovers, or verify a remainder used in an algorithm. The underlying idea is part of modular arithmetic as described by Wolfram MathWorld.
How to calculate
- The calculator opens with the demonstration 2,026 mod 7, so finite results and a validated example Excel workbook are available immediately.
- Replace Dividend (x) with the integer you want to divide. Replace Divisor (y) with a positive, nonzero integer.
- Read Remainder (r) first. Then use Integer quotient (q), Largest multiple, Divisibility, and the identity line to audit the result.
- Select Download Excel to create a fresh workbook from the currently validated controls. It contains Summary, Inputs, Calculation, and Notes sheets.
- Select Reset to clear the demonstration and all calculated content. Reset disables Download Excel until both fields again contain a complete valid state.
Input guide
Dividend (x) is required and must be a safe whole number. Enter digits with an optional leading plus or minus sign; standard comma grouping such as 2,026 is accepted. Decimals, scientific notation, spaces inside the number, and decimal-comma forms such as 1,5 are rejected rather than silently reinterpreted. A larger dividend usually increases the quotient, while the remainder cycles between zero and one less than the divisor.
Divisor (y) is also required. It must be a positive safe integer greater than zero, for example 7. A zero divisor is undefined, and a negative divisor introduces sign-convention differences that this calculator intentionally avoids. Increasing the divisor widens the possible remainder range from 0 through y – 1; it does not guarantee that the actual remainder rises.
Output guide
Remainder (r) is the exact amount left after complete groups of the divisor are removed. Zero means the dividend is divisible by the divisor. Integer quotient (q) is the floor quotient used in Euclidean division. Largest multiple equals q × y. Divisibility reports Yes only when the remainder is zero. The three summary pills repeat the current expression, legal remainder range, and divisibility state. The Calculation detail table lists Dividend, Divisor, Integer quotient, Largest multiple, Remainder, and Reconstructed dividend; the final row must exactly equal the input dividend.
Worked example
For the startup values, divide 2,026 by 7. The largest whole-number quotient is 289 because 289 × 7 = 2,023. Subtracting gives 2,026 – 2,023 = 3. Therefore 2,026 mod 7 = 3, the divisibility result is No, and the identity is 2,026 = 289 × 7 + 3. The first-open cards, table, live summary, and workbook all use these same exact values.
How the modulo formula works
For an integer dividend x and a positive divisor y, Euclidean division gives unique integers q and r such that x = q × y + r and 0 ≤ r < y. The calculator computes q = floor(x ÷ y), then r = x – q × y. This remains non-negative even when the dividend is negative, as long as the divisor is positive.
Modulo is useful for periodic systems: a weekday index repeats every 7 positions, a 12-hour clock repeats every 12, and array positions can wrap after a fixed length. Congruence notation says two integers have the same remainder for a chosen modulus. The Encyclopedia of Mathematics entry on congruence gives the broader number-theory context.
Modulo versus programming-language remainder
Positive integers behave consistently across ordinary arithmetic and most programming languages. Differences become important when a negative operand is involved. Python's modulo result follows the sign of the divisor, as documented in the Python expression reference. JavaScript's percent operator is formally a remainder operator whose result follows the sign of the dividend; the MDN remainder operator guide explains that distinction.