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

2 Answers
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*3*5*7*11=2310#

Apr 30, 2018

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#