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# If x is a positive integer, is x/21 an integer?

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Math Expert
Joined: 02 Sep 2009
Posts: 51215
If x is a positive integer, is x/21 an integer?  [#permalink]

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26 Jun 2018, 04:51
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Difficulty:

55% (hard)

Question Stats:

60% (01:31) correct 40% (01:17) wrong based on 25 sessions

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If x is a positive integer, is $$\frac{x}{21}$$ an integer?

(1) $$\frac{4x}{21}$$ is an integer.

(2) $$\frac{12x}{21}$$ is an integer.

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Re: If x is a positive integer, is x/21 an integer?  [#permalink]

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26 Jun 2018, 05:12
Bunuel wrote:
If x is a positive integer, is $$\frac{x}{21}$$ an integer?

(1) $$\frac{4x}{21}$$ is an integer.

(2) $$\frac{12x}{21}$$ is an integer.

Factorize 21, we get 3 * 7. So the numerator x should have at-least one 3 and one 7 then$$\frac{x}{21}$$will be an integer.

Option 1 :

$$\frac{4x}{21}$$ --> 4 = $$2^2.$$
For $$\frac{4x}{21}$$to be an integer, x should have at-least one 3 and one 7.

So, $$\frac{x}{21}$$ is an integer. Option 1 satisfies.

Option 2:

$$\frac{12x}{21}$$ ------> 12 = $$2^2 * 3$$.
Here we already have 3, so $$x = 7$$ will make $$\frac{12x}{21}$$ a integer.
But x=7, doesn't make \frac{x}{21} a integer. So definite NO.

We need to prove definite YES.
$$\frac{12x}{21}$$------> $$x = 21$$ , will make $$\frac{12x}{21}$$an integer
So, x/21 is also an integer for x=21. So definite YES.

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Re: If x is a positive integer, is x/21 an integer?  [#permalink]

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26 Jun 2018, 06:21

Solution

Given:
• x is a positive integer

To find:
• is $$\frac{x}{21}$$ an integer

Analysing Statement 1
• As per the information given in statement 1, $$\frac{4x}{21}$$ is an integer
• As 4 and 21 are co-prime to each other, x must be divisible by 21
o Therefore, $$\frac{x}{21}$$ must be an integer

Hence, statement 1 is sufficient to answer

Analysing Statement 2
• As per the information given in statement 2, $$\frac{12x}{21}$$ is an integer
• The numbers 12 and 21 are not co-prime to each other, hence, we can’t say whether x is divisible by 21 or not

Hence, statement 2 is not sufficient to answer

Hence, the correct answer is option A.

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Re: If x is a positive integer, is x/21 an integer? &nbs [#permalink] 26 Jun 2018, 06:21
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