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# If x and y are integers, then

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If x and y are integers, then [#permalink]  07 Aug 2017, 13:03
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Question Stats:

64% (02:36) correct 35% (01:29) wrong based on 73 sessions

If x and y are integers, then $$\frac{(x) (x + 1) (x + 2)}{2*3*5^y}$$ must be an integer if which of the following is true ?

A. x is even.

B. x is odd.

C. x is divisible by three.

D. y is even.

E. y is equal to zero.
[Reveal] Spoiler: OA

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Re: If x and y are integers, then [#permalink]  07 Feb 2018, 01:03
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In here, the numerator is the product of consecutive integers. So it would be divisible by 2 and 3. If y=0, then the given fraction will be an integer.
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Re: If x and y are integers, then [#permalink]  07 Feb 2018, 01:47
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take $$y = 10$$ and for consecutive integers take some small numbers such as $$1, 2, 3 or 2, 3, 4 or 3, 4, 5$$
doing so will invalidate all of the options from A to D because non of the final result will be an integer
Leaving only E for ans
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Re: If x and y are integers, then [#permalink]  09 May 2018, 08:00
amorphous wrote:
take $$y = 10$$ and for consecutive integers take some small numbers such as $$1, 2, 3 or 2, 3, 4 or 3, 4, 5$$
doing so will invalidate all of the options from A to D because non of the final result will be an integer
Leaving only E for ans

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Re: If x and y are integers, then [#permalink]  09 May 2018, 09:23
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When we assume $$x, x+1$$ and $$x+2$$ as $$1,2,3$$ or $$2,3,4$$ or $$3,4,5$$
while replacing these values in the given expression we can cancel of digits.
In the first two cases $$2$$ and $$3$$ are cancelled off and we are left with $$\frac{1}{5^y}$$ and $$\frac{4}{5^y}$$
Take $$y$$ as a large number because we are working with a $$must be$$ type questions so the ans must be valid in all circumstances.
In both the case the ans will not be an integer unless y = 0.
For the final case $$3$$ gets cancelled of and we are left with $$\frac{10}{5^y}$$. For this we can still get an integer if $$y$$= odd i.e $$1$$ but it is not the case always as we have found out earlier.
therefore best way to answer this question is to chose a large value for $$y$$ and make sure we can eliminate as many options as possible.
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Re: If x and y are integers, then [#permalink]  07 Jul 2018, 10:42
amorphous wrote:
When we assume $$x, x+1$$ and $$x+2$$ as $$1,2,3$$ or $$2,3,4$$ or $$3,4,5$$
while replacing these values in the given expression we can cancel of digits.
In the first two cases $$2$$ and $$3$$ are cancelled off and we are left with $$\frac{1}{5^y}$$ and $$\frac{4}{5^y}$$
Take $$y$$ as a large number because we are working with a $$must be$$ type questions so the ans must be valid in all circumstances.
In both the case the ans will not be an integer unless y = 0.
For the final case $$3$$ gets cancelled of and we are left with $$\frac{10}{5^y}$$. For this we can still get an integer if $$y$$= odd i.e $$1$$ but it is not the case always as we have found out earlier.
therefore best way to answer this question is to chose a large value for $$y$$ and make sure we can eliminate as many options as possible.

So for this types of questions, there is no general rule? It's trial and error approach?
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Re: If x and y are integers, then [#permalink]  07 Jul 2018, 11:13
Expert's post
Sometimes the only way to tackle a question is trial and error, even though is time consuming.

However, amorphous above provided a pretty fast and elegant solution to you

Quote:
take $$y = 10$$ and for consecutive integers take some small numbers such as $$1, 2, 3 or 2, 3, 4 or 3, 4, 5$$
doing so will invalidate all of the options from A to D because non of the final result will be an integer
Leaving the only E for ans

To speed up, even more, the process you could think about two things:

- first off: when a question asks you "which of the following" start always from the bottom of the answer choices. More often than not the correct answer is from the bottom-up:
- secondly: the numerator of the fraction is 3 consecutive integers.

if x is even = 2 AND y= 0, which means that $$5^0 = 1$$ then you do have

$$\frac{2*3*4}{2*3*1}$$

2 and 3 cancel out and you do have $$\frac{4}{1} = 4$$.

Now, the question has only one correct answer and it is also a must be true question. As such, E must be the correct answer without bothering you to check the others.

Hope now is clear to you.

Regards
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Re: If x and y are integers, then   [#permalink] 07 Jul 2018, 11:13
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