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For a positive integer p, the index-3 of p is defined as the greatest

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For a positive integer p, the index-3 of p is defined as the greatest  [#permalink]

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New post 21 Sep 2018, 01:32
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For a positive integer p, the index-3 of p is defined as the greatest integer n such that 3^n is factor of p. For example, the index-3 of 54 is 3 as 3 is the greatest exponent of 3 and is a factor of 54. If q and r are positive integers, is the index-3 of q greater than the index-3 of r ?

(1) q − r > 0
(2) q/r is a multiple of 3

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Re: For a positive integer p, the index-3 of p is defined as the greatest  [#permalink]

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New post 21 Sep 2018, 02:07
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Bunuel wrote:
For a positive integer p, the index-3 of p is defined as the greatest integer n such that 3^n is factor of p. For example, the index-3 of 54 is 3 as 3 is the greatest exponent of 3 and is a factor of 54. If q and r are positive integers, is the index-3 of q greater than the index-3 of r ?

(1) q − r > 0
(2) q/r is a multiple of 3


To know if the index-3 of q is greater than the index-3 of p, we need to know if the prime factorization of q has a larger power of 3 than the prime factorization of p/
We'll look for statements that give us this information, a Logical approach.

(1) This tells us nothing about the prime factorization of p and q.
Insufficient.

(2) Since q/r is an integer, then all the factors of r are factors of q, and since q/r is a multiple of 3, q has at least one 3 in its prime factorization in addition to all those that r have. This is exactly what we need! (as it means that the 3-index of q is larger than that of r)
Sufficient.

(B) is our answer.
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Re: For a positive integer p, the index-3 of p is defined as the greatest &nbs [#permalink] 21 Sep 2018, 02:07
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