STEP I-2009-1

A proper factor of an integer N is a positive integer, not 1 or N, that divides N.

Show that 3^2 \times 5^3 has exactly 10 proper factors. Determine how many other integers of the form 3^m \times 5^n (where m and n are integers) have exactly 10 proper factors.

Let N be the smallest positive integer that has exactly 426 proper factors. Determine N, giving your answer in terms of its prime factors.

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