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# M31-33

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Math Expert
Joined: 02 Sep 2009
Posts: 52296

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14 Jun 2015, 13:39
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Difficulty:

85% (hard)

Question Stats:

35% (02:21) correct 65% (01:37) wrong based on 46 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

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14 Jun 2015, 13:39
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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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.
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Joined: 02 Sep 2009
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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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08 Jan 2019, 00:18
can you please elaborate on statement 1.How can a divisor be more than a dividend and still leave a remainder?
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08 Jan 2019, 00:29
Dannys wrote:
can you please elaborate on statement 1.How can a divisor be more than a dividend and still leave a remainder?

Let me ask you a question: how many leftover apples would you have if you had 60 apples and wanted to distribute in 120 baskets evenly? Each basket would get 0 apples and 120 apples would be leftover (remainder).

So, when a divisor is more than dividend, then the remainder equals to the dividend, for example:
3 divided by 4 yields the reminder of 3: $$3=4*0+3$$;
9 divided by 14 yields the reminder of 9: $$9=14*0+9$$;
1 divided by 9 yields the reminder of 1: $$1=9*0+1$$.

For more on this check:

5. Divisibility/Multiples/Factors

6. Remainders

For other subjects:
ALL YOU NEED FOR QUANT ! ! !
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08 Jan 2019, 00:46
Bunuel wrote:
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

Given x +ive Integer
123/x gives Remainder =3

x can be any of these values 6,40,60,120

Statement 1
Remainder when 60/x is > or = 60

This is only possible when x =120, because if you take other values from the sample set,
60/40, Remainder = 20
60/6, Remainder= 0
60/60, Remainder= 0
60/120, Remainder= 60

Statement 2
x can be 60,120,240
Will give 2 values for x, 60 and 120.

Only Statement 1 is sufficient for answering the question

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If you notice any discrepancy in my reasoning, please let me know. Lets improve together.

Quote which i can relate to.
Many of life's failures happen with people who do not realize how close they were to success when they gave up.

Re: M31-33 &nbs [#permalink] 08 Jan 2019, 00:46
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# M31-33

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