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Modular Multiplication Calculator

Enter a, b and a modulus m to compute (a × b) mod m.

Result

4
56,088
7

How it is calculated

The tool computes (a × b) mod m: the remainder when the product of a and b is divided by m. The result is always between 0 and m − 1, even when a negative number is involved, so (−7 × 5) mod 12 gives 1, not −11 (the product is −35 and the non-negative remainder is 1). All three inputs must be whole numbers, and the modulus must be at least 1; a decimal such as 2.5, or m = 0 or a negative m, shows no result. With m = 1 the answer is always 0. Each input can be at most 253 = 9,007,199,254,740,992 in size; a typed integer even one above that, such as 9007199254740993, is rejected and shows no result. The product is printed exactly in full, digit by digit, and the remainder is found without overflow, so even two numbers near that limit give a correct remainder and a product with more than 30 digits. The tool shows the result, the exact product and the modulus.

Worked example

Compute (123 × 456) mod 7. Multiply: 123 × 456 = 56,088. Divide by 7: 7 × 8,012 = 56,084, which leaves 56,088 − 56,084 = 4. A shortcut gives the same answer: 123 mod 7 = 4 (since 7 × 17 = 119) and 456 mod 7 = 1 (since 7 × 65 = 455), and 4 × 1 = 4. The product and the modulus are shown next to the result: the product was 56,088 and m was 7.

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Common mistakes

  • Expecting a negative result for a negative product. This tool returns the non-negative remainder, so (−7 × 5) mod 12 is 1, not −11 as in some programming languages.
  • Entering a decimal. Only whole numbers are accepted, so 2.5 shows no result instead of being rounded.
  • Using m = 0 or a negative modulus. The modulus must be at least 1; with m = 1 the result is always 0.
  • Typing numbers larger than 253 (9,007,199,254,740,992), such as the digits of an RSA modulus. They are rejected, even by a single unit (9007199254740993 gives no result); reduce them first or use a big-integer tool.
  • Confusing the order of operations with (a mod m) × b. They agree only after you take the mod of the whole product at the end, as this tool does.

About this calculator

Modular multiplication computes (a × b) mod m — first multiply the two integers normally, then take the remainder after dividing by the modulus m. It behaves like ordinary multiplication but 'wraps around' once the product passes a multiple of m, the same way a 12-hour clock wraps 13 o'clock back to 1.

This operation is a core building block of modern cryptography: RSA encryption and decryption are, at their core, repeated modular multiplications carried out on very large numbers, and Diffie-Hellman key exchange relies on the same operation to make key derivation easy to compute forward but hard to reverse. It also underlies hash functions, pseudorandom number generators, and check-digit schemes like the ISBN or credit card Luhn check, and it is a standard early topic in discrete mathematics and number theory courses because it introduces students to arithmetic that does not behave like ordinary integer arithmetic.

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