Review question

# What's the smallest number with four different prime factors? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R7162

## Solution

$f(x)$ denotes the number of different prime factors of the integer $x$, excluding $1$ and $x$ itself. For example, since $12=2\times2\times3$, then $f(12)=2$.

1. Calculate $f(121)$.

$121=11\times 11$. So it has only one repeated prime factor, so $f(121)=1$.

1. Calculate $f(17)$.

Since $17$ is prime, that is, $17=1\times 17$, and we are meant to exclude $1$ and $17$, we must have that $f(17)=0$.

1. Write down the smallest $x$ for which $f(x)=4$.

The smallest $x$ for which $f(x)=4$ will be the product of the four smallest prime numbers. So $x=2\times3\times5\times7=210$.