Combinatorics | Number Theory Answers

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. 

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. 

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. 

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. 

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

4. Prove that for positive integer n we have 169|3³ⁿ⁺³-26n-27. 

3. Prove that there exists infinitely many positive integers n such that 4n²+ 1 is divisible both by 5 and 13. 

2. Find all integers x #= 3 such that x-3Ix³-3.

1. Find all positive integers n such that n²+ 1 is divisible by n+ 1. 

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

a)6 798 340

b)54 786 978