Review question

# What's the smallest integer with 426 proper factors? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7465

## Question

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

1. 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.

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