Number Theory Solver

The Number Theory Solver covers number theory and the discrete mathematics that surrounds it. GCD and LCM with the Euclidean algorithm written out, and the extended algorithm for Bezout coefficients. Prime factorisation and divisibility tests. Modular arithmetic: reduce, invert, solve linear congruences, apply the Chinese remainder theorem, compute large powers by repeated squaring, and apply Fermat's little theorem and Euler's theorem. Combinatorics: permutations, combinations, the pigeonhole principle, inclusion-exclusion, and counting strings and arrangements. Proofs by induction, contradiction, contrapositive, and cases. Set operations, relations, functions, and recurrence relations. Logic questions, truth tables, equivalences, and quantifier negation, and graph theory basics, such as degree sums, Euler paths, and simple tree properties, are handled in the same style.

Type the question or photograph it: "find gcd(252, 198)", "solve 7x = 3 mod 26", "how many 5-letter strings have no repeated letter", "prove by induction that the sum of the first n odd numbers is n^2". For a proof, state the method the course expects. For a counting problem, say whether repetition is allowed and whether order matters. Set Answer form to exact; results here are integers and fractions. For a modular problem, always include the modulus. For a set problem, write the sets with braces or describe them in words. For a recurrence, give the recurrence and the initial terms and say whether you want a closed form, the next terms, or a proof about it.

Output: the number, the residue, the count, or the finished proof first. Then the algorithm steps, each division or reduction on its own line, or the proof written with the base case, the inductive hypothesis, and the inductive step labeled. Counting problems state the model before the formula. The working is shown so you can check it. For probability built on a count, use the Probability Solver. For algorithms and code that implement these ideas, use the Computer Science Solver. A follow-up can ask for a proof rewritten by another method, for the count confirmed by a different argument, or for the algorithm shown on new numbers. The working stays at the level of a discrete-math course, with each rule of inference or counting principle named.