Modulo Calculator – Remainder of Division

The modulo operation reports what remains after taking out whole multiples of the divisor. 17 mod 5 = 2 because 17 = 3×5 + 2; the quotient 3 is discarded and the remainder 2 is the answer. Clock arithmetic is everyday modulo: 22:00 is 10 pm because 22 mod 12 = 10, and adding 7 + 9 hours lands at 16 → 4 o’clock since 16 mod 12 = 4. The calculator handles negative dividends with the mathematical convention, −7 mod 3 = 2 (because −7 = −3×3 + 2), and flags the programming convention (−1) so you can match your language. Evenness testing is the classic filter — n mod 2 = 0 means even, and 100 mod 7 = 2 shows how remainders detect non-divisibility. Congruence notation a ≡ b (mod m) appears too: 17 ≡ 2 (mod 5), the basis of modular arithmetic.

1. 1

   Enter dividend and divisor

   Values like 17 and 5

2. 2

   Divide and drop the quotient

   17 ÷ 5 = 3 remainder 2; only the remainder survives

3. 3

   Read the remainder

   17 mod 5 = 2

4. 4

   Interpret the congruence

   17 ≡ 2 (mod 5) links the result to clock-style arithmetic

• 1

  a mod b ∈ {0, 1, …, b−1} under the math convention

• 2

  mod 2 = 0 tests even; mod 3 = 0 tests divisibility by three

• 3

  Programming languages may return −1 for −7 mod 3; math says 2

• Confusing the remainder with the quotient (3 vs 2 for 17 mod 5)

• Expecting −7 mod 3 to be negative; mathematical result is +2

• Computing 100 mod 7 as 14 — it is 2 (14×7 = 98)

• Using modulo with a divisor of 0, which is undefined