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# If a and n are integers, and a^3=360n, then n must be divisible by

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Joined: 02 Sep 2009
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If a and n are integers, and a^3=360n, then n must be divisible by  [#permalink]

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24 Oct 2016, 21:56
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35% (medium)

Question Stats:

67% (01:03) correct 33% (01:29) wrong based on 112 sessions

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If a and n are integers, and a^3=360n, then n must be divisible by which of the following?

A. 2
B. 6
C. 25
D. 27
E. 60

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Re: If a and n are integers, and a^3=360n, then n must be divisible by  [#permalink]

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24 Oct 2016, 23:20
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1
Bunuel wrote:
If a and n are integers, and a^3=360n, then n must be divisible by which of the following?

A. 2
B. 6
C. 25
D. 27
E. 60

a^3=360n = 2^3*3^2*5*n

and a^3 is a perfect cube due to a being an integer

hence 360n must have powers of each prime number as multiple of 3

hence 360n = 2^3*3^2*5*(3*5^2) for least value of n

i.e. n must have 5^2

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Re: If a and n are integers, and a^3=360n, then n must be divisible by  [#permalink]

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25 Oct 2016, 06:42
Bunuel wrote:
If a and n are integers, and a^3=360n, then n must be divisible by which of the following?

A. 2
B. 6
C. 25
D. 27
E. 60

$$a^3=360n$$

Or, $$a^3$$ = $$2^3$$ x $$3^2$$ x $$5^1$$ x $$n$$

Hence $$n$$ = $$3$$ x $$5^2$$

Thus among the given options only (C) satisfies...

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Re: If a and n are integers, and a^3=360n, then n must be divisible by  [#permalink]

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25 Oct 2016, 16:02
1
Bunuel wrote:
If a and n are integers, and a^3=360n, then n must be divisible by which of the following?

A. 2
B. 6
C. 25
D. 27
E. 60

We are given that a^3=360n, which means that 360n is a perfect cube. We must remember that all perfect cubes break down to unique prime factors, each of which has an exponent which is a multiple of 3. So let’s break down 360 into primes to help determine the minimum value of y.

360 = 36 x 10 = 6 x 6 x 2 x 5 = 2 x 3 x 2 x 3 x 2 x 5 = 2^3 x 3^2 x 5^1

In order to make 360n a perfect cube, the smallest value of n is 3^1 x 5^2 so that 360n = (2^3 x 3^2 x 5^1) x (3^1 x 5^2) = 2^3 x 3^3 x 5^3, which is a perfect cube.

Now that we have the minimum value of n we can determine which answer choice MUST divide into n.

A) 2

(3^1 x 5^2)/2 DOES NOT equal an integer

B) 6

(3^1 x 5^2)/6 DOES NOT equal an integer

C) 25

(3^1 x 5^2)/25 DOES equal an integer

D) 27

(3^1 x 5^2)/27 DOES NOT equal an integer

E) 60

(3^1 x 5^2)/60 DOES NOT equal an integer

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Re: If a and n are integers, and a^3=360n, then n must be divisible by  [#permalink]

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20 Dec 2018, 08:09
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Re: If a and n are integers, and a^3=360n, then n must be divisible by &nbs [#permalink] 20 Dec 2018, 08:09
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