Number Theory

Problems related to integers, prime numbers, congruences, etc.

Problems in this category

We are asked to prove by contrapositive that if $n$ is an integer and $3n+7$ is odd, then $n$ is eve...

Proof by ContrapositiveInteger PropertiesOdd and Even NumbersModular Arithmetic (Implied)

The problem asks us to use a direct proof to show that the cube of every even integer is even.

Number TheoryEven IntegersProofsInteger PropertiesDirect Proof

The problem asks to find the greatest common factor (G.C.F.) of the numbers 30, 45, and 60, given th...

Greatest Common FactorGCDPrime FactorizationInteger Properties

The problem asks us to find which of the given numbers (11,116, 16,462, 21,240, 28,814) is divisible...

Divisibility RulesInteger PropertiesDivisibility by 2Divisibility by 3Divisibility by 6

We are given that $x$, $y$, and $z$ are natural numbers such that $1 < x < y < z$, and we need to fi...

Diophantine EquationsInequalitiesInteger Solutions

The problem asks us to find all triples of natural numbers $(x, y, z)$ such that $1 < x < y < z$ and...

Diophantine EquationsInequalitiesNumber Theory

We are given that $x, y, z$ are natural numbers such that $1 < x < y < z$ and $(1 + \frac{1}{x})(1 +...

Diophantine EquationsInequalitiesInteger SolutionsNumber Theory

We are given the HCF (highest common factor) and LCM (least common multiple) of two numbers, and one...

HCFLCMNumber Properties

The problem consists of filling in the blanks in sentences related to number theory.

Number PropertiesEven NumbersOdd NumbersInteger Properties

We are asked to find $\varphi(n)$ for $n \le 310$ and we are given $n = 4$. It is presumed we have t...

Euler's Totient FunctionNumber TheoryPrime Factorization

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