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# M31-33

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Math Expert
Joined: 02 Sep 2009
Posts: 43792

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14 Jun 2015, 13:39
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Difficulty:

75% (hard)

Question Stats:

39% (01:50) correct 61% (01:38) wrong based on 31 sessions

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If $$x$$ is a positive integer and 123 divided by $$x$$ leaves a remainder of 3, what is the value of $$x$$?

(1) The remainder when 60 is divided by $$x$$ is more than or equal to 60.

(2) $$x$$ is a multiple of 60
[Reveal] Spoiler: OA

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Math Expert
Joined: 02 Sep 2009
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14 Jun 2015, 13:39
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Official Solution:

If $$x$$ is a positive integer and 123 divided by $$x$$ leaves a remainder of 3, what is the value of $$x$$?

The stem says that 123 is 3 more than a multiple of 3: $$123 = xq + 3$$, which gives $$120= xq$$. Therefore $$x$$ must be a factor of 120 greater than 3 (divisor, $$x$$, must be more than the remainder 3). So, $$x$$ can be: 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60 or 120.

(1) The remainder when 60 is divided by $$x$$ is more than or equal to 60. The remainder, cannot be greater than the dividend, thus the remainder remainder when 60 is divided by $$x$$ IS 60. Which implies that $$x$$ is greater than 60. Hence $$x = 120$$. Sufficient.

(2) $$x$$ is a multiple of 60. From above $$x$$ can be 60 or 120. Not sufficient.

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Joined: 11 Apr 2016
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Location: India
Concentration: Marketing, Technology

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29 Mar 2017, 05:35
Hi Brunel,

Can you please elaborate on why x must be more than the remainder 3 ? I was not able to understand the same.
Math Expert
Joined: 02 Sep 2009
Posts: 43792

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29 Mar 2017, 05:51
varunjoshi31 wrote:
Hi Brunel,

Can you please elaborate on why x must be more than the remainder 3 ? I was not able to understand the same.

Because the remainder (3 in our case) is always less than divisor (x in our case).

Check the links below for more:
Theory on remainders problems
Tips on remainders

DS remainders problems
PS remainders problems

Hope it helps.
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Re: M31-33   [#permalink] 29 Mar 2017, 05:51
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# M31-33

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