Divisibility and gcd
94 problems · Divisibility rules, greatest common divisor, Euclid's algorithm and Bezout's identity.
Ex.204Basic divisibility and linear combinations
Divisibility and gcd
High school
Ex.205Consecutive integers are coprime
Divisibility and gcd
High school
Ex.206Finding divisors systematically
Divisibility and gcd
High school
Ex.232Divisibility rules
Divisibility and gcd
High school
Ex.233GCD of a GCD
Divisibility and gcd
High school
Ex.234The HCF of consecutive integers
Divisibility and gcd
High school
Ex.325The Euclidean algorithm and Bézout
Divisibility and gcd
L1
Ex.3346 divides n³ − n
Divisibility and gcd
High school
Ex.366Common divisors and common multiples
Divisibility and gcd
High school
Ex.367Computing LCM from divisibility
Divisibility and gcd
High school
Ex.368$n^2-1$ is divisible by 8 for odd $n$
Divisibility and gcd
High school
Ex.369The relation between divisibility and lcm
Divisibility and gcd
High school
Ex.370Simplifying a fraction with the divisibility
Divisibility and gcd
High school
Ex.371divisibility of three numbers
Divisibility and gcd
High school
Ex.453Coprime integers
Divisibility and gcd
High school
Ex.454GCD of three numbers
Divisibility and gcd
High school
Ex.455Congruence classes modulo $n$
Divisibility and gcd
L1
Ex.456GCD and divisibility
Divisibility and gcd
High school
Ex.457LCM of three numbers
Divisibility and gcd
High school
Ex.458The least common multiple
Divisibility and gcd
High school
Ex.459Every integer has a prime divisor
Divisibility and gcd
High school
Ex.620Counting divisors and the hundred lockers
Divisibility and gcd
Olympiad
Ex.623gcd(n! + 1, (n + 1)! + 1) = 1
Divisibility and gcd
M1
Ex.62413 divides 2⁷⁰ + 3⁷⁰
Divisibility and gcd
M1
Ex.645The sum of φ(d) over the divisors of n
Divisibility and gcd
M1
Ex.70630 divides n⁵ − n
Divisibility and gcd
Olympiad
Ex.740A divisibility identity for a sum
Divisibility and gcd
L1
Ex.741Bézout identity for 1001 and 315
Divisibility and gcd
High school
Ex.742When are $2n$ and $3n+1$ coprime?
Divisibility and gcd
High school
Ex.743Coprime powers remain coprime
Divisibility and gcd
L1
Ex.744If $a,b$ are coprime, so are $a+b$ and $ab$
Divisibility and gcd
L1
Ex.745Divisibility of $a^n-b^n$
Divisibility and gcd
L1
Ex.746Divisors of a product
Divisibility and gcd
L1
Ex.747Why the Euclidean algorithm works
Divisibility and gcd
L1
Ex.748The Euclidean algorithm and Bézout identity
Divisibility and gcd
High school
Ex.749When is $(6n+12)/(2n+1)$ an integer?
Divisibility and gcd
High school
Ex.751When does $n-4$ divide $3n-17$?
Divisibility and gcd
High school
Ex.752When is the number of divisors odd?
Divisibility and gcd
L1
Ex.753A divisibility recurrence
Divisibility and gcd
L1
Ex.754The smallest prime divisor
Divisibility and gcd
L1
Ex.908Bézout's identity
Divisibility and gcd
High school
Ex.909Coprime product
Divisibility and gcd
L1
Ex.910Infinitely many primes
Divisibility and gcd
High school
Ex.911The fundamental theorem of arithmetic
Divisibility and gcd
High school
Ex.912Gauss's lemma
Divisibility and gcd
L1
Ex.913GCD and product of coprime factors
Divisibility and gcd
L1
Ex.914Divisors of coprime numbers
Divisibility and gcd
L1
Ex.915Linear relations and coprimality
Divisibility and gcd
L1
Ex.916Proof of the Euclidean algorithm
Divisibility and gcd
High school
Ex.917GCD as a linear combination
Divisibility and gcd
High school
Ex.918GCD and linear combinations
Divisibility and gcd
L1
Ex.919GCD of powers
Divisibility and gcd
L1
Ex.920A coprimality proof
Divisibility and gcd
L1
Ex.921A recursive GCD
Divisibility and gcd
L1
Ex.922Semiprimes and divisors
Divisibility and gcd
L1
Ex.923Additive property of the p-adic valuation
Divisibility and gcd
Olympiad
Ex.924Finding modular inverses
Divisibility and gcd
L1
Ex.925Number of divisors
Divisibility and gcd
L1
Ex.926Perfect numbers
Divisibility and gcd
Olympiad
Ex.927Perfect squares and divisors
Divisibility and gcd
L1
Ex.928Prime power divisibility
Divisibility and gcd
L1
Ex.929The radical of an integer
Divisibility and gcd
Olympiad
Ex.930Sum of divisors
Divisibility and gcd
L1
Ex.931Units modulo $n$
Divisibility and gcd
L1
Ex.932p-adic valuation
Divisibility and gcd
L1
Ex.1277Long runs of non-squarefree numbers
Divisibility and gcd
Olympiad
Ex.1280The gcd of 2ᵃ − 1 and 2ᵇ − 1
Divisibility and gcd
Olympiad
Ex.128160 divides xyz for every Pythagorean triple
Divisibility and gcd
Olympiad
Ex.1329(n²)! is divisible by (n!)^(n+1)
Divisibility and gcd
Olympiad
Ex.1331Infinitely many n dividing 2ⁿ + 1
Divisibility and gcd
Olympiad
Ex.1333v_p(n!) from the digits of n in base p
Divisibility and gcd
Olympiad
Ex.1482Existence of a Bézout identity
Divisibility and gcd
L1
Ex.1484A common divisor divides 43
Divisibility and gcd
High school
Ex.1487Solving $\divisibility(a,b)+\operatorname{lcm}(a,b)=9+b$
Divisibility and gcd
High school
Ex.1488A prime divisor property
Divisibility and gcd
L1
Ex.1644Egyptian fractions
Divisibility and gcd
Olympiad
Ex.1645Chinese remainder theorem
Divisibility and gcd
L1
Ex.1646Coprime numbers and exponents
Divisibility and gcd
Olympiad
Ex.1647Coprime sums and products
Divisibility and gcd
Olympiad
Ex.1648An application of the Euclidean algorithm
Divisibility and gcd
Olympiad
Ex.1649A Diophantine equation with coefficients
Divisibility and gcd
L1
Ex.1650Euler's totient function
Divisibility and gcd
L1
Ex.1651GCD of sums
Divisibility and gcd
Olympiad
Ex.1652GCD via the Euclidean matrix
Divisibility and gcd
L1
Ex.1653Divisors of Mersenne numbers
Divisibility and gcd
Olympiad
Ex.1654GCD in $\mathbb{Z}$ vs $\mathbb{R}[X]$
Divisibility and gcd
Olympiad
Ex.1655GCD of polynomial coefficients
Divisibility and gcd
Olympiad
Ex.1656Primitive Pythagorean triples
Divisibility and gcd
Olympiad
Ex.1957⌊√n + √(n+1)⌋ = ⌊√(4n+2)⌋
Divisibility and gcd
Olympiad
Ex.1961Prime factors of Fermat numbers
Divisibility and gcd
Olympiad
Ex.1995Even perfect numbers (Euclid–Euler)
Divisibility and gcd
Olympiad
Ex.1999gcd(F_m, F_n) = F_gcd(m,n)
Divisibility and gcd
Olympiad
Ex.2122Carmichael numbers (statement)
Divisibility and gcd
M1
Ex.2123The Miller–Rabin test
Divisibility and gcd
M1
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