Answer:
The answer would be 30
Step-by-step explanation
For positive integers: 2 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 = 2310 For all integers: ± ( 2 ⋅ 3 ⋅ 5 ) = ± 30 For Gaussian integers: ± 1 ± 3 i 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 + 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: ( 1 + i ) ( 1 + 2 i ) = − 1 + 3 i , 1 + 3 i , − 1 − 3 i , − 1 + 3 i , 3 + i , 3 − i , − 3 + i , − 3 − i
(which just means 3 times the square root of 7).