SOLUTION: The product of all the ages of the teenagers at a party was 2 971 987 200. The number of 18 year olds who attended the party?

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Question 1126866: The product of all the ages of the teenagers at a party was 2 971 987 200. The number of 18 year olds who attended the party?
Found 3 solutions by addingup, ikleyn, greenestamps:
Answer by addingup(3677)   (Show Source): You can put this solution on YOUR website!
did you divide? try it: 2971987200/18
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
.
2 971 987 200 = .


Now you need to combine / (to distribute) the factors to get the product of natural numbers, each of which is between 13 and 18.


The only way to do it is THIS :


              =  = 13 * (7*2)^2 * (5*3)^2 * (9*2)^2 * (2^4) = 13*14*14*15*15*18*18*16.


Answer.  One is 13 years old;

         2 of 14 years old;

         2 of 15 years old;

         2 of 18 years old;

         1 of 16 years old.


Check.   13 * 14*14* 15*15 * 18*18 * 16 = (my MS Excel) = 2 971 987 200.

Solved.

======================

The tutor @greenestamps put this line in his post 

    I think the other tutor completely missed the point of the problem....

If it relates to my post, then I do not understand the meaning of his remark.

My post came first.

It contains the complete and correct solution to the problem.

@greenestamps saw my post and then repeated practically the same solution.

After that, he states that I missed something . . .


@greneestamps, I sincerely ask you to edit your post IMMEDIATELY to avoid any misunderstanding.

.

It would be also right to bring your apologies.



Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Cool problem!

I think the other tutor completely missed the point of the problem....

We need to find the prime factorization of the number and then put the prime factors together so that all the ages are in the range 13-19.

The prime factorization is



(1) Clearly there must be exactly one 13-year-old.

The remaining prime factors are

(2) The only multiple of 7 in the required range is 14; since there are 2 factors of 7, there must be 2 14-year-olds. That uses both factors of 7 and 2 of the factors of 2.

The remaining prime factors are

(3) The only multiple of 5 in the required range is 15; since there are 2 factors of 5, there must be 2 15-year olds. That uses both factors of 5 and 2 of the factors of 3.

The remaining prime factors are

(4) The only age in the required range that is a multiple of 3 and has only 2 and 3 as prime factors is 18; that age requires 1 factor of 2 and 2 factors of 3. Since there are 4 factors of 3 remaining, there must be 2 18-year-olds. That uses all 4 factors of 3 and 2 of the remaining 6 factors of 2.

The remaining prime factors are

So there is 1 16-year-old.

SOLVED!

The ages of the teenagers at the party:
13: 1
14: 2
15: 2
16: 1
18: 2

CHECK:

Answer to the question that was asked: there are two 18-year-olds at the party.
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