Question 731432
THE ANSWER:
{{{highlight(1+2+4+5+6=18)}}} {{{highlight(1*2*4*5*6=240)}}}
 
HOW TO GET TO THE ANSWER:
We need to find factors of 240.
Either systematically or sloppily.
Systematically:
{{{240=2^4*3*5}}} prime factorization that can be obtained by dividing over and over by prime numbers in order, starting from 2 and going up:
240 ÷ {{{red(2)}}} = 120
120 ÷ {{{red(2)}}} = 60
60 ÷ {{{red(2)}}} = 30
30 ÷ {{{red(2)}}} = 15
15 ÷ {{{red(3)}}} = {{{red(5)}}}
So 240 = {{{red(2)}}}X{{{red(2)}}}X{{{red(2)}}}X{{{red(2)}}}X{{{red(3)}}}X{{{red(5)}}}
The sloppy way:
I know that 240 = 30 X 8, so
{{{24=8*30=8*10*3=8*2*5*3=2*2*2*2*5*3}}}
 
Then, it's a question of turning the 6 factors in {{{2*2*2*2*3*5}}} 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+5=8}}} and the four remaining 2's cannot be grouped into the three additional different whole numbers.
Using {{{2*3=6}}} and {{{2*5=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(1)}}} 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(2)}}} and {{{2*2=blue(4)}}}
What do we do with {{{3}}} and {{{5}}}?
{{{2*3=blue(6)}}} and {{{blue(5)}}} as two of the five different whole numbers is an obvious choice, because using {{{2*5=10}}} and {{{3}}} would make the sum too large:
{{{1+3+10+2+4=14+2+4=20}}}.
So {{{highlight(blue(1))}}}, {{{highlight(blue(2))}}}, {{{highlight(blue(4))}}}, {{{highlight(blue(5))}}}, and {{{highlight(blue(6))}}} are the five different whole numbers that multiply to yield 240,
{{{highlight(1*2*4*5*6=240)}}},
and
{{{highlight(1+2+4+5+6=18)}}} is the 5-number sum.