# Whats the smallest composite number that has the five smallest prime numbers as factors?

Apr 30, 2018

See explanation.

#### Explanation:

The number which has five smallest prime numbers as factors would be the product of the prime numbers:

## $n = 2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 = 2310$

Apr 30, 2018

For positive integers: $2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 = 2310$

For all integers: $\pm \left(2 \cdot 3 \cdot 5\right) = \pm 30$

For Gaussian integers: $\pm 1 \pm 3 i$ and $\pm 3 \pm i$ (all combinations of signs)

#### Explanation:

A prime number is a number whose only factors are itself, units and unit multiples of itself.

So in the positive integers, the first few primes are:

$2 , 3 , 5 , 7 , 11 , \ldots$

So the smallest composite positive integer with the five smallest prime positive integers as factors is:

$2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 = 2310$

If we extend our interest to include negative integers, then the smallest primes are:

$2 , - 2 , 3 , - 3 , 5 , - 5 , \ldots$

So the smallest composite integers with the five smallest prime integers as factors are:

$\pm \left(2 \cdot 3 \cdot 5\right) = \pm 30$

If we consider Gaussian integers, then the smallest primes are:

$1 + i$, $1 - i$, $- 1 + i$, $- 1 - i$, $1 + 2 i$, $1 - 2 i$, $- 1 + 2 i$, $- 1 - 2 i$, $2 + i$, $2 - i$, $- 2 + i$, $- 2 - i$, $3$, $- 3$,...

So the smallest composite Gaussian integers with the five smallest prime Gaussian integers as factor are:

$\left(1 + i\right) \left(1 + 2 i\right) = - 1 + 3 i$, $1 + 3 i$, $- 1 - 3 i$, $- 1 + 3 i$, $3 + i$, $3 - i$, $- 3 + i$, $- 3 - i$