Question

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

Answer 1

See explanation.

Explanation:

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

`n=2*3*5*7*11=2310`

Answer 2

For positive integers: `2 * 3 * 5 * 7 * 11 = 2310`

For all integers: `+-(2 * 3 * 5) = +-30`

For Gaussian integers: `+-1+-3i` and `+-3+-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,...`

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

`2 * 3 * 5 * 7 * 11 = 2310`

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

`2, -2, 3, -3, 5, -5,...`

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

`+-(2 * 3 * 5) = +-30`

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

`1+i`, `1-i`, `-1+i`, `-1-i`, `1+2i`, `1-2i`, `-1+2i`, `-1-2i`, `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:

`(1+i)(1+2i) = -1+3i`, `1+3i`, `-1-3i`, `-1+3i`, `3+i`, `3-i`, `-3+i`, `-3-i`