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Re: Positive integers a and b each have exactly four factors. If
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15 Aug 2018, 05:21
Explanation
Start by considering integer a, which is the most constrained variable. It is a positive one-digit number (between 1 and 9, inclusive) and it has four factors. Prime numbers have exactly two factors:
themselves and one, so only look at non-prime one-digit positive integers. That’s a short enough list:
1 has just one factor
4 has three factors: 1, 2, and 4
6 has four factors: 1, 2, 3, and 6
8 has four factors: 1, 2, 4, and 8
9 has three factors: 1, 3, and 9
So the two possibilities for a are 6 and 8. Now apply the two constraints for b. It is 9 greater than a, and it has exactly four factors. Check the possibilities:
If a = 6, then b = 15, which has four factors: 1, 3, 5, and 15.
If a = 8, then b = 17, which is prime, so it has only has two factors: 1 and 17.
Only b = 15 works, so a must be 6.