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Combinatorics | Number Theory

45* . Prove that if a, b, c are three different integers, then there exist in-

finitely many positive integers n such that a+n, b+n, c+n are pairwise rel-

atively prime.

Combinatorics | Number Theory

43. Prove that there exists an increasing infinite sequence of tetrahedral

numbers (i.e. numbers of the form T_{n} = 1/6 n(n+ 1)(n+2), n = 1,2, ... ), such

that every two of them are relatively prime.

Combinatorics | Number Theory

42. Prove that there exists an increasing infinite sequence of triangular

numbers (i.e. numbers of the form t_{n} = -1/2 n(n+ 1), n = 1, 2, ... ) such that

every two of them are relatively prime.

Combinatorics | Number Theory

41. Prove that for every integer k the numbers 2k+1 and 9k+4 are rel-

atively prime, and for numbers 2k-1 and 9k+4 find their greatest common

divisor as a function of k.

Combinatorics | Number Theory

5. Prove that "19|2^{2^{6k+2}}+3" for k = 0, 1, 2, ....

Combinatorics | Number Theory

4. Prove that for positive integer n we have 169|3^{3n+3}-26n-27.

Combinatorics | Number Theory

3. Prove that there exists infinitely many positive integers n such that 4n^{2}+ 1 is divisible both by 5 and 13.

Combinatorics | Number Theory

2. Find all integers x #= 3 such that x-3Ix^{3}-3.

Combinatorics | Number Theory

1. Find all positive integers n such that n^{2}+ 1 is divisible by n+ 1.

Combinatorics | Number Theory

Test the following numbers for divisibility by 6,9,11 (do not divide or factorise)

a)6 798 340

b)54 786 978