Homework Answers

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. 

The number of traffic violation recorded by police department for 10 day period is given. Calculate the first quartile, second quartile, third quartile and inter quartile range.

22 19 25 24 18 15 9 12 16 20

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.