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

##### 2 Answers

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#

For positive integers:

For all integers:

For Gaussian integers:

#### 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#