SOLUTION: What five different whole numbers add up to 18 multiply those same numbers and get 240?

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Question 731432: What five different whole numbers add up to 18 multiply those same numbers and get 240?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
THE ANSWER:
highlight%281%2B2%2B4%2B5%2B6=18%29 highlight%281%2A2%2A4%2A5%2A6=240%29

HOW TO GET TO THE ANSWER:
We need to find factors of 240.
Either systematically or sloppily.
Systematically:
240=2%5E4%2A3%2A5 prime factorization that can be obtained by dividing over and over by prime numbers in order, starting from 2 and going up:
240 ÷ red%282%29 = 120
120 ÷ red%282%29 = 60
60 ÷ red%282%29 = 30
30 ÷ red%282%29 = 15
15 ÷ red%283%29 = red%285%29
So 240 = red%282%29Xred%282%29Xred%282%29Xred%282%29Xred%283%29Xred%285%29
The sloppy way:
I know that 240 = 30 X 8, so
24=8%2A30=8%2A10%2A3=8%2A2%2A5%2A3=2%2A2%2A2%2A2%2A5%2A3

Then, it's a question of turning the 6 factors in 2%2A2%2A2%2A2%2A3%2A5 into five factors by grouping (multiplying) some together, and maybe including a 1 as a factor.
If we do not include a 1, to add up to the even number 18, we could have both odd factors (3 and 5) by themselves, adding to 12, or both paired with a 2, to become even numbers.
Neither choice works.
With 3 and 5 as two of the five different whole numbers, we have
3%2B5=8 and the four remaining 2's cannot be grouped into the three additional different whole numbers.
Using 2%2A3=6 and 2%2A5=10 as two of the five different whole numbers is not possible either, because we would just have two extra 2's to make three additional whole numbers from.
With blue%281%29 as one of the five different whole numbers, what would we do with 3 and 5?
We need one and only one additional odd number.
The other odd number must be turned into an even number by grouping it with a 2.
We would be left with three 2's to be turned into the two additional different whole numbers.
Those two additional different whole numbers can only be blue%282%29 and 2%2A2=blue%284%29
What do we do with 3 and 5?
2%2A3=blue%286%29 and blue%285%29 as two of the five different whole numbers is an obvious choice, because using 2%2A5=10 and 3 would make the sum too large:
1%2B3%2B10%2B2%2B4=14%2B2%2B4=20.
So highlight%28blue%281%29%29, highlight%28blue%282%29%29, highlight%28blue%284%29%29, highlight%28blue%285%29%29, and highlight%28blue%286%29%29 are the five different whole numbers that multiply to yield 240,
highlight%281%2A2%2A4%2A5%2A6=240%29,
and
highlight%281%2B2%2B4%2B5%2B6=18%29 is the 5-number sum.