Number theory

Use divisibility and congruences to explain integer-solution conditions and construct proofs or counterexamples.

Meaning

Number theory studies divisibility, primes, remainders, and integer solutions. Congruence notation concisely expresses that two integers leave the same remainder under a specified divisor.

Example

means that 17 minus 5 is a multiple of 12. A 12-hour clock, where 17 hours later reaches the same position as 5 hours later, provides a model.Modulo 4, 2 and 6 are congruent, but dividing both by 2 gives 1 and 3, which are not. Whether cancellation preserves a congruence depends on common-factor conditions with the modulus.Having the same remainder does not mean two integers are equal. Division in a congruence cannot be performed by any arbitrary number as if it were an ordinary equation.

Practice for this concept

Scope and references

Topics

Primes, congruences, integer solutions, Diophantine equations

Learning goal

Use divisibility and congruences to explain integer-solution conditions and construct proofs or counterexamples.

Helpful background

Explore the divisors and multiples, primes, equations, and proofs needed for the chosen topic.

Congruences and complete integer solutions

These tasks do not assess general theorems about primes or higher-degree integer equations.

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