1. Prove that fraction (m(n+1)+1)/(m(n+1)-n) is irreducible for every positive integer’s m and n.
2. Do there exist 10 distinct integers such that the sum of any 9 of them is a perfect square?
Yes, for example 61, 56, 53, 48, 37, 13, -2, -28, -59, -108. Infect one can find infinitely many such sets.
3. Prove that any integer a ≥ 7 is the sum of two relatively prime integers. (we say two integers m and n are relatively if their HCF (i.e. GCD) is 1)
When a is odd
a = p + q where either p is even or q is even
If p is even the q is odd and hence p and q are relatively prime to each other. Similarly vice versa. i. e. GCD (p, q) = 1
Thus any integer n ≥ 7 is the sum of two relatively prime integers.
5. Find the smallest natural number of the form 2a3b7c, such that the half of the number is the cube of an integer, one third of the number is the seventh power of an integer and one seventh of the number is square of an integer.
[Hint: (2a3b7c)/2 is a cube only if a – 1, b, c is all divisible by 3].
Given (2a3b7c)/2 = n3 => 2(a-1)3b7c = n3
(2a3b7c)/3 = m3 => 2(a)3(b-1)7c = m7
(2a3b7c)/7 = l2 => 2(a)3b7(c-1) = l2
By trial and error method we find that
a = 28 b = 36 and c = 21
Furthermore a – 1, b, c are divisible by 3