<PRE><B>Question 1126866</B>
&lt;br&gt;
Cool problem!&lt;br&gt;
I think the other tutor completely missed the point of the problem....&lt;br&gt;
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.&lt;br&gt;
The prime factorization is&lt;br&gt;
{{{2971987200 = (2^8)(3^6)(5^2)(7^2)(13)}}}&lt;br&gt;
(1) Clearly there must be exactly one 13-year-old.&lt;br&gt;
The remaining prime factors are {{{(2^8)(3^6)(5^2)(7^2)}}}&lt;br&gt;
(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.&lt;br&gt;
The remaining prime factors are {{{(2^6)(3^6)(5^2)}}}&lt;br&gt;
(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.&lt;br&gt;
The remaining prime factors are {{{(2^6)(3^4)}}}&lt;br&gt;
(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.&lt;br&gt;
The remaining prime factors are {{{2^4=16}}}&lt;br&gt;
So there is 1 16-year-old.&lt;br&gt;
SOLVED!&lt;br&gt;
The ages of the teenagers at the party:
13: 1
14: 2
15: 2
16: 1
18: 2&lt;br&gt;
CHECK: {{{(13)(14^2)(15^2)(16)(18^2) = 2871987200}}}&lt;br&gt;
Answer to the question that was asked: there are two 18-year-olds at the party.</PRE>