Factors — XAT Previous-Year Questions

We have two unknown positive integers m and n, whose product is less than 100.
There are two additional statement of facts available:

1. mn is divisible by six consecutive integers { j, j + 1,...,j + 5 }
2. m + n is a perfect square.

Which of the two statements above, alone or in combination shall be sufficient to determine the numbers m and n?

If N = (11^p+7)(7^q-2)(5^r+1)(3^s) is a perfect cube, where p, q, r and s are positive integers, then the smallest value of p + q + r + s is:

For two positive integers a and b, if (a + b)^(a+b) is divisible by 500, then the least possible value of a × b is: