A 3-digit number is the product of 4 prime numbers. Given that the 3 digits of the number are all prime & different and that the sum of ...its prime factors is 30, find the number.
The prime digits are 2,3,5 and 7.
So we can form only 4! numbers which are different 3 digit prime numbers. So our choice number is from of these 24 numbers. In fact these could be physically tried. But a short cut is always better. So we proceed as shown below:
Since the factors of the number are 4 prime numbers, which add up to 30, we should have the factors 2,3,5,7,11,13,17,19,23 and 29.
Examining the different primes from 2 to 29 , no 4 different primes add up to 30.
However , the conditions given say only different prime 3 digits. But the condition does not say the 4 factors should be different primes.
If the 3 digit prime number allows for 4 prime factors , not neccesarily different, then 2*2*7*19 = 532 is a solution.
Number of factors are four .
Factors are: 2, 2, 7 and 19.
All are prime factors.
Sum of the factors: 2+2+7+19 = 30.
The digits of 532 are 5, 3 and 2 are three distinct primes.
Hope this helps.