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# Is the positive integer x a multiple of 16?

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Is the positive integer x a multiple of 16?  [#permalink]

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10 Dec 2017, 23:42
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Difficulty:

55% (hard)

Question Stats:

62% (01:38) correct 38% (01:28) wrong based on 88 sessions

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Is the positive integer x a multiple of 16?

(1) x is a multiple of 8.
(2) x^2 is a multiple of 32.

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Re: Is the positive integer x a multiple of 16?  [#permalink]

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11 Dec 2017, 00:06
S1: x can be 8 or 16: not sufficient.
S2: x^2 is multiple of 32: 2^5. that means x can be 8 or 16. Not sufficient.
Even combination of statement 1&2 doesn't give any new info.
Hence insufficient.

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Re: Is the positive integer x a multiple of 16?  [#permalink]

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11 Dec 2017, 00:15
S1: x can be 8 or 16: not sufficient.
S2: x^2 is multiple of 32: 2^5. that means x can be 8 or 16. Not sufficient.
Even combination of statement 1&2 doesn't give any new info.
Hence insufficient.

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Re: Is the positive integer x a multiple of 16?  [#permalink]

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11 Dec 2017, 04:26
1
Bunuel wrote:
Is the positive integer x a multiple of 16?

(1) x is a multiple of 8.
(2) x^2 is a multiple of 32.

For a number to be divisible by 16, that number must have at least four 2's in its prime factorisation (because 16 = 2^4).

(1) x is a multiple of 8, or of 2^3. This guarantees that x has at least three 2's, but we cant say about four 2's. Insufficient.

(2) x^2 is a multiple of 32, or of 2^5. Now since x is a positive integer, x^2 is a perfect square, which means all powers of prime numbers in prime factorisation of x^2 will be even. So x^2 cannot just have five 2's, if its a multiple of 32 (2^5), it will have at least six 2's.(x^2 will be at-least 2^6). This in turn tells us that x will be at least 2^3, or we can say that x is a multiple of 8. Same as first statement. Insufficient.

Since both statements tell us the same thing, even after combining them we cant say whether x has more than three 2's or not. So Insufficient.

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Re: Is the positive integer x a multiple of 16?  [#permalink]

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17 Apr 2018, 07:27
+1 for option E.

St 1 - NS
St 2 - NS ; x is of the form 8k. (bçoz - if x^2 =32k ; x will have one factor of 4 and one factor of 2 to be a perfect square).

Both together - NS
The answer is option E
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Re: Is the positive integer x a multiple of 16?   [#permalink] 17 Apr 2018, 07:27
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# Is the positive integer x a multiple of 16?

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