What is the easiest way to solve a problem like this:
k is a positive integer and 225 and 216 are both divisors of k.
If k=2^a x 3^b x 5^c, where a,b,and c are positive integers, what is the least possible value of a+b+c?
(A) 4
(B) 5
© 6
(D) 7
(E) 8
So right off the bat you can eliminate (A) and (B). Must you find the least common multiple of 216 and 225, then factor that? Is there a quicker way to do this? Thanks in advance.
225=3^2 x 5^2
216=2^3 x 3^3
So k must have 2^3 x 3^3 x 5^2 as a factor.
So the least possible value of a+b+c is 3+3+2 = 8.
Break 225 and 216 down to prime factors:
3x3x5x5
2x2x2x3x3x3
The smallest set of prime factors that includes both of those is
2x2x2x3x3x3x5x5
That’s three 2s, three 3s, two 5s, for a total of eight.
Yes. No.
On preview: What they said.
Ah, thanks
Note that, in doing what they did, Giles and Ximenean did find the least common multiple (in factored form—they didn’t find the l.c.m. and then factor it).