Applications of Congruence
Divide a, b, and n by their common divisor, invert the reduced coefficient, find one residue, then add steps of the reduced modulus. To continue from Congruence, try coefficient inverse.
Math calculator
Congruence Calculator
Solve ax ≡ b (mod n) and list every distinct solution modulo n. The result panel keeps solution residues and its numerical trail together.
Congruence inputs
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Checking Congruence on one case
For 14x ≡ 30 mod 100, gcd(14,100)=2 divides 30. Reducing gives 7x ≡15 mod50, whose solutions modulo 100 are 45 and 95.
The mathematical idea behind Congruence
Start the Congruence review with Coefficient a. Compare Coefficient a with its source, then test Target b in a second Congruence run without changing the first Congruence case.
Conditions that alter Congruence
The modulus must be positive. If the GCF does not divide b, no integer x can satisfy the congruence.
Read Solution residues against Coefficient a, not in isolation. Use Target b to estimate the Congruence magnitude. When Positive modulus is altered, label the new Congruence trial. Its Solution residues should not replace the original Congruence answer.
From Congruence output to working record
Adding n to any solution produces the same residue class. That behavior gives the congruence output a built-in reasonableness test.
A modular inverse solves the special coprime step; this page also handles multiple or absent solutions. Writing “Solution residues” beside the output prevents that mix-up.
Why several solutions appear
When g=gcd(a,n) divides b, reducing by g creates one solution modulo n/g. Viewed in the original modulus n, that solution repeats g times at intervals of n/g. Listing all standard residues prevents a correct but incomplete single answer from hiding the other classes.
Congruences model repeating schedules, divisibility constraints, cyclic positions, and modular equations. A related application of Congruence is remainder convention.
A linear congruence has solutions when gcd(a,n) divides b. When it does, the number of distinct residues equals that GCF. Congruence also connects with coprime condition.
Cross-checking Congruence
Units and conventions belong with a Congruence answer. Confirm that Coefficient a and Target b use the intended interpretation, then label Solution residues the same way. A numerically correct Solution residues can still answer the wrong Congruence question when that context changes.
Questions about Congruence
Can there be more than one residue?
Yes, when gcd(a,n)>1 divides b.
When is there no solution?
When gcd(a,n) does not divide b.
Why list residues only up to n−1?
Every other solution is congruent to one of them.