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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.
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