(n + 1)(n + 2)

If n is a natural number,
 prove that the number (n + 1)(n + 2) ................. (n + 10) is not a perfect square.

m + n is divisible by 24

Let m and n be natural numbers and let mn + 1 be divisible by 24.
Show that m + n is divisible by 24.

divisible by 12

Let a, b, c, d be integers. 
Show that the product (a - b)(a - c)(a - d)(b - c)(b - d)(c - d) is divisible by 12.

among any ten consecutive positive integers

Prove that among any ten consecutive positive integers at least one is relatively prime to the
product of the others.

ab + 1, bc + 1, and ca + 1 are perfect squares

Find in finitely many triples (a, b, c) of positive integers such that a, b, c are
in arithmetic progression and such that ab + 1, bc + 1, and ca + 1 are perfect squares.

xy + 1, yz + 1, zx + 1 are all perfect squares

Show that if x, y, z are positive integers, then (xy + 1)(yz + 1)(zx + 1) is a perfect square if
and only if xy + 1, yz + 1, zx + 1 are all perfect squares.

number of integral roots of p(x)

Let  be a polynomial with integral coefficients. Let a , b , c  be three distinct integers such that  
p(a) = p(b) = p(c) = - 1. Find the number of integral roots of p(x). 

the number of polynomials p(x)

Find the number of polynomials p(x) with integral coefficients satisfying
the conditions p(1)=2, p(3)=1.