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# What is the remainder when the positive integer x is divided

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What is the remainder when the positive integer x is divided  [#permalink]

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21 Jan 2012, 17:24
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75% (01:44) correct 25% (01:48) wrong based on 880 sessions

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What is the remainder when the positive integer x is divided by 6?

(1) When x is divided by 2, the remainder is 1; and when x is divided by 3, the remainder is 0
(2) When x is divided by 12, the remainder is 3.

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21 Jan 2012, 17:52
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enigma123 wrote:
What is the remainder when the positive integer x is divided by 6?
1). When x is divided by 2, the remainder is 1; and when x is divided by 3, the remainder is 0
2). When x is divided by 12, the remainder is 3.

What is the remainder when the positive integer x is divided by 6?

This question can be very easily solved with plug-in method:

(1) When x is divided by 2, the remainder is 1 --> x is an odd number AND "when x is divided by 3, the remainder is 0" --> x is a multiple of 3 --> so, x is an odd multiple of 3: 3, 9, 15, 21, ... --> you can see a definite pattern here that any such number divided by 6 yields remainder of 3. Sufficient.

(2) When x is divided by 12, the remainder is 3 --> x is of a type $$x=12q+3$$: 3, 15, 27, ... --> any such number divided by 6 yields remainder of 3 (or you can notice that 12q is divisible by 6 and 3 divided by 6 yields the remainder of 3). Sufficient.

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Re: What is the remainder when the positive integer x is divided  [#permalink]

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11 Aug 2013, 21:55
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Easiest it can be:

REM(X/6)?

(1).

X = 2A + 1 ...... X can be 1,3,5,7...
X = 3B .......X can be 3,6,9,...

Combined series X = LCM of (2,3) + common term in the series

X = 6C + 3 hence remainder is '3' SUFFICIENT

(2).

X = 12D + 3
X = 6(2D) + 3

Hence remainder is '3' SUFFICIENT

So (D) it is !
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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08 Jan 2016, 00:27
Is there an algebraic approach to this problem? i am always confused whether to take an algebraic approach or number testing approach esp on remainder prblems. I don't want to make this decision in the exam hall. If I want to go in the exam hall with one approach which one it should be for remainder problems? Thanks
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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08 Jan 2016, 01:11
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NoHalfMeasures wrote:
Is there an algebraic approach to this problem? i am always confused whether to take an algebraic approach or number testing approach esp on remainder prblems. I don't want to make this decision in the exam hall. If I want to go in the exam hall with one approach which one it should be for remainder problems? Thanks

Hi,
each Q may have different method to be tackled efficiently ..
But the statements generally give you sufficient info to find the answer..
Algebric way may be better if you are to find the numeric value of remainder, may be in PS..
and working on the info avail in terms of putting values etc in case we are to find if there would be any remainder, but value is not required..

here the info is very straightforward..

What is the remainder when the positive integer x is divided by 6?
1). When x is divided by 2, the remainder is 1; and when x is divided by 3, the remainder is 0
it is not div by 2, so will not be div by 6... suff

2). When x is divided by 12, the remainder is 3.
since div by 12 leaves an odd remainder, x is an odd number but 6 is an even number.. again suff
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1) Absolute modulus : http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
2)Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html
3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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What is the remainder when the positive integer x is divided  [#permalink]

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08 Jan 2016, 01:19
however the question does not ask if x is divisible by 6? it asks what is the remainder when x is divided by 6. St2 is quite straight forward. we can clearly see how the remainder should be 3. But st1 is not as intuitively clear. any thoughts there?
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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08 Jan 2016, 01:32
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NoHalfMeasures wrote:
however the question does not ask if x is divisible by 6? it asks what is the remainder when x is divided by 6. St2 is quite straight forward. we can clearly see how the remainder should be 3. But st1 is not as intuitively clear. any thoughts there?

Hi,
sorry , i did not read the Q properly..
Statement 1 is slightly complex and we require to again play with the propperties of number..

three things..
1) 3 has alternate odd and even multiple..
2) we know x is an odd number, since it is not div by 2..
3) from 1 and 2 above x is an odd multiple of 3..
4) Also all even multiple of 3 will be div of 3, so the diff in x and lower even multiple of 3 will be 3, ..
hence remainder is 3
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3) effects of arithmetic operations : https://gmatclub.com/forum/effects-of-arithmetic-operations-on-fractions-269413.html

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What is the remainder when the positive integer x is divided  [#permalink]

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25 Oct 2016, 07:54
Hello,

I still don't understand why statement (1) is sufficient.

The number 3 fulfills all the requirements of statement (1). Hence, when it is divided by 6, there will be no remainder.
Like already mentioned before all the other numbers e.g. 9, 15, ... will result in a remainder of 3 when it is divided by 6.
My conclusion would be that there will be two possible solutions, either 0 or 3 --> Insufficient

Thank you very much for your help!
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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25 Oct 2016, 08:09
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MaximilianKrempl wrote:
Hello,

I still don't understand why statement (1) is sufficient.

The number 3 fulfills all the requirements of statement (1). Hence, when it is divided by 6, there will be no remainder.
Like already mentioned before all the other numbers e.g. 9, 15, ... will result in a remainder of 3 when it is divided by 6.
My conclusion would be that there will be two possible solutions, either 0 or 3 --> Insufficient

Thank you very much for your help!

3 divided by 6 gives the remainder of 3: 3 = 0*6 + 3.
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What is the remainder when the positive integer x is divided  [#permalink]

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15 Jul 2017, 14:12
NoHalfMeasures wrote:
Is there an algebraic approach to this problem? i am always confused whether to take an algebraic approach or number testing approach esp on remainder prblems. I don't want to make this decision in the exam hall. If I want to go in the exam hall with one approach which one it should be for remainder problems? Thanks

If you're a high-achiever, then I would just take a couple hours, head to YouTube, and learn the basics of modular arithmetic. Then be good to go on all remainder problems. True, you don't NEED modular arithmetic for the GMAT and it will never be tested directly, but it sure is useful. Why not give yourself a structured, clean, and systematic way of handling these types of problems rather than piddling around with what should be simple stuff? Back in the day, they used to teach this in elementary schools in the US. They probably still do in countries with better math education. So it's not difficult.

In this case, we're given that:

$$x ≡ r (mod 6)$$

Statement A tells us:

$$x ≡ 1 (mod 2)$$
$$x ≡ 0 (mod 3)$$

This is clearly sufficient, since we're asked to find a value in $$mod 6$$, and the $$LCM$$ of $$2$$ and $$3$$ is $$6$$.

But if we really want to solve it, it's quick to do. We get "$$x ≡ 3 (mod 6)$$"

So we know the remainder is 3.

Sufficient.

Statement B tells us:

$$x ≡ 3 (mod 12)$$

Then it would also be true that in mod 6, the remainder is 3.

Sufficient.
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Re: What is the remainder when the positive integer x is divided  [#permalink]

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31 Jul 2018, 06:40
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