Answer:
6, 9
Explanation:
You want the numbers that have a product of 54 and a sum of 15.
Divisors of 54
The prime factors of 54 are ...
54 = 2 · 3³
Adding 1 to the exponents, we have (1, 3) +(1, 1) = (2, 4). The product of these increased exponents is 2·4 = 8, which means there are 8 divisors of 54, including 1 and 54.
We can write the factor pairs as ...
54 = 1·54 = 2·27 = 3·18 = 6·9
The sums of these divisor pairs are 55, 29, 21, 15
The two numbers you seek are 6 and 9.
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Additional comment
The reason for figuring the number of divisors is so we can check to make sure we have found factor pairs for all of them. The 8 divisors give us 8/4 = 4 factor pairs. We have found it is usually easier to find the desired sum once we have the list of factor pairs for a number.
The smallest sum of two factors will be 2√54 ≈ 14.7. This tells you that the sum of 15 will involve factors that are close to √54 ≈ 7.4. You find the factors of interest when you consider divisors that are near 7: 6 and 9.
You may have noticed it is very helpful to have a great command of your multiplication facts when working problems of this sort.
A graph can provide a quick solution, too.
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