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k is a positive integer and 225 and 216 are both divisors of

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k is a positive integer and 225 and 216 are both divisors of [#permalink]

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k is a positive integer and 225 and 216 are both divisors of k. If k=(2^a)*(3^b)*(5^c), where a, b and c are positive integers, what is the least possible value of a+ b+ c?

A. 4
B. 5
C. 6
D. 7
E. 8
[Reveal] Spoiler: OA

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Re: k is a positive integer and 225 and 216 are both divisors of [#permalink]

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vix wrote:
k is a positive integer and 225 and 216 are both divisors of k. If k=(2^a)*(3^b)*(5^c) , where a, b and c are positive integers, what is the least possible value of a+ b+ c?

A) 4
(B) 5
(C) 6
(D) 7
(E) 8

Please help me with solution of this problem.


Make prime factorization of 225 and 216: 225=15^2=3^2*5^2 and 216=6^3=2^3*3^3.

Now, in order to minimize the sum of the powers of k's primes, we should minimize k itself. Minimum value of k will be the least common multiple of 225 and 216 (since k is a multiple of both): LCM(225, 216)=k=2^3*3^3*5^2 --> a+b+c=3+3+2=8.

Answer: E.

For more in this check Number Theory Chapter of Math Book: math-number-theory-88376.html

Hope it helps.
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Re: k is a positive integer and 225 and 216 are both divisors of [#permalink]

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Lets make factorization of 225 and 216 ..

225 = 5 X 5 X 3 X 3 X 3

216 = 2 X 2 X 2 X 3 X 3 X 3

K would have to have 3 two's , 225 has 3 threes and and so does 216 but they can be the same three threes so we count them only once ... 225 has 2 fives ... So we had them together and we get 3 + 3 + 2 = 8 (answer) ...
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Re: k is a positive integer and 225 and 216 are both divisors of [#permalink]

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Re: k is a positive integer and 225 and 216 are both divisors of   [#permalink] 12 Nov 2017, 08:16
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