Modular Inverse Calculator – a⁻¹ mod m

The modular inverse of a is the number that multiplies it to 1 modulo m. Since 3×5 = 15 = 2×7+1, we write 3⁻¹ ≡ 5 (mod 7); similarly 2⁻¹ ≡ 3 (mod 5) because 6 = 5+1. The calculator uses the extended Euclidean algorithm and shows every step: for 3 mod 7, back-substituting 7 = 2×3+1 gives 1 = 7−2×3, so −2 ≡ 5 is the inverse. Existence is not guaranteed — the inverse exists only when gcd(a,m) = 1, so 4 has no inverse modulo 8 (gcd 4) while 3 does modulo 8 (3⁻¹ = 3, since 9 ≡ 1). Applications anchor cryptography: RSA decryption exponents are modular inverses of the encryption exponent mod φ(n), and affine ciphers decode through a⁻¹ mod 26. The tool also verifies the result by multiplying back and reducing.

1. 1

   Enter a and modulus m

   Such as 3 and 7

2. 2

   Check coprimality

   gcd(a,m) must equal 1 or no inverse exists

3. 3

   Run extended Euclid

   Bézout coefficients expose the inverse

4. 4

   Verify by multiplication

   a × a⁻¹ mod m must reduce to 1

• 1

  No inverse unless gcd(a,m) = 1

• 2

  The inverse, if present, is unique within 1…m−1

• 3

  Self-inverses exist: 3×3 ≡ 1 (mod 8)

• Assuming every a has an inverse — 4 fails modulo 8

• Reporting the raw Bézout coefficient when it is negative; add m

• Confusing a⁻¹ mod m with 1/a in the rationals

• Forgetting to verify a×a⁻¹ ≡ 1 (mod m) at the end