Free · Modular arithmetic
Signed Modular Inverses and Coprimality
Find additive or multiplicative residues for signed integer inputs and verify coprimality.
- Formula & worked example
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Calculator inputs
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How to use this calculator
- Enter the known values in the units shown. Results update as you type.
- Where results are editable, change one to solve backwards. Lock a value to hold it fixed.
- Use the worked example to check the method. Reset restores the starting fields.
Private by default
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Formula and method
The multiplicative inverse of 3 modulo 11 is 4.
a × x ≡1 (mod m) or a + x ≡0 (mod m)
Worked example
Enter these known values and leave the other values blank.
- a
- 3
- m
- 11
- Result
- For a = 3.
- Explanation
- a × x ≡ 1 (mod 11).
- Explanation
- Least nonnegative solution residue x = 4.
- Explanation
- Verification: 3 × 4 = 12; 12 ≡ 1 (mod 11).
Assumptions and limitations
- The integer a is nonzero and its magnitude is at most1e10; m is a positive integer at most1e10.
- A multiplicative inverse exists only when gcd(a,m)=1. Noncoprime pairs retain the explicit no-solution explanation; a Bezout coefficient is not presented as a valid inverse.
- Additive mode returns the least nonnegative residue of−a. Modulus1 uses the captured residue0 convention.
Common questions
Can I enter a negative value?
Yes for a, in either inverse mode. The modulus remains positive.
What is the output range?
The selected residue lies from zero through one less than the modulus.
References
- Exact integer arithmetic reference
- Exact integer arithmetic reference
- Calculation definition and unit reference
Calculate additive and multiplicative inverses for signed nonzero integers, check coprimality and show the complete equation. Correct the signed inverse: −3 modulo 11 has multiplicative inverse 7 because −3 × 7 ≡ 1 modulo 11. Report when no multiplicative inverse exists and enforce the stated integer limits.