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Intern  Joined: 14 Jul 2010
Posts: 6
What is the remainder when a is divided by 4?  [#permalink]

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7 00:00

Difficulty:   5% (low)

Question Stats: 79% (01:04) correct 21% (01:36) wrong based on 255 sessions

### HideShow timer Statistics What is the remainder when a is divided by 4?

(1) a is the square of an odd integer.
(2) a is a multiple of 3.

According to the book, the answer is A.

But according to me, the answer should be C since if we take the first statement and use the value 1 for a, it gives us 1 for the square of 1, what would be the remainder?
When we use 3 for a, the square of a will give us 9 which when divided by 4, gives us remainder of 1.

Thanks.
Kash.
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Joined: 02 Sep 2009
Posts: 56373
Re: MGMAT's Number Properties Data Sufficiency Question  [#permalink]

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ksear wrote:
Hi everyone,

I'm a little confused regarding the correct answer for the below mentioned question:

What is the remainder when a is divided by 4?

(1) a is the square of an odd integer.
(2) a is a multiple of 3.

According to the book, the answer is A.

But according to me, the answer should be C since if we take the first statement and use the value 1 for a, it gives us 1 for the square of 1, what would be the remainder?
When we use 3 for a, the square of a will give us 9 which when divided by 4, gives us remainder of 1.

Thanks.
Kash.

Positive integer $$a$$ divided by positive integer $$d$$ yields a reminder of $$r$$ can always be expressed as $$a=qd+r$$, where $$q$$ is called a quotient and $$r$$ is called a remainder, note here that $$0\leq{r}<d$$ (remainder is non-negative integer and always less than divisor).

So according to above, when positive integer $$a$$ is less than divisor $$d$$ then remainder upon division $$a$$ by $$d$$ is always equals to $$a$$, for example 5 divided by 10 yields reminder of 5. So when 1 is divided by 4 remainder is 1.

Or algebraically: 1 divided by 4 can be expressed as $$1=0*4+1$$, so $$r=1$$.

Back to the original question:

What is the remainder when a is divided by 4?

(1) a is the square of an odd integer --> $$a=(2k+1)^2=4k^2+4k+1$$, first two terms (4k^2 and 4k) are divisible by 4 and the third term (1) when divided by 4 yields the remainder of 1. Sufficient.

Or you can try several numbers for $$a$$:
$$a=1^1=1$$ --> 1 divided by 4 yields remainder of 1;
$$a=3^1=9$$ --> 9 divided by 4 yields remainder of 1;
$$a=5^1=25$$ --> 25 divided by 4 yields remainder of 1;
...

(2) a is a multiple of 3 --> clearly insufficient as $$a$$ can as well be a multiple of 4, 12 for example, and in this case the remainder will be 0, and it also can not be a multiple of 4, 3 for example, and in this case the remainder will be 3. Not sufficient.

Questions on remainders:

PS:
remainder-101074.html
remainder-problem-92629.html
number-properties-question-from-qr-2nd-edition-ps-96030.html
remainder-when-k-96127.html
ps-0-to-50-inclusive-remainder-76984.html
good-problem-90442.html
remainder-of-89470.html
number-system-60282.html
remainder-problem-88102.html

DS:
remainder-problem-101740.html
remainder-101663.html
ds-gcd-of-numbers-101360.html
data-sufficiency-with-remainder-98529.html
sum-of-remainders-99943.html
ds8-93971.html
need-solution-98567.html
gmat-prep-ds-remainder-96366.html
gmat-prep-ds-93364.html
ds-from-gmatprep-96712.html
remainder-problem-divisible-by-86839.html
gmat-prep-2-remainder-86155.html
remainder-94472.html
remainder-problem-84967.html

Hope it helps.
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Re: MGMAT's Number Properties Data Sufficiency Question  [#permalink]

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ksear wrote:
Hi everyone,

I'm a little confused regarding the correct answer for the below mentioned question:

What is the remainder when a is divided by 4?

(1) a is the square of an odd integer.
(2) a is a multiple of 3.

According to the book, the answer is A.

But according to me, the answer should be C since if we take the first statement and use the value 1 for a, it gives us 1 for the square of 1, what would be the remainder?
When we use 3 for a, the square of a will give us 9 which when divided by 4, gives us remainder of 1.

Thanks.
Kash.

To answer your specific question, if you divide 1 by 4, the reminder is 1. So the answer is valid.
Intern  Joined: 14 Jul 2010
Posts: 6
Re: MGMAT's Number Properties Data Sufficiency Question  [#permalink]

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It does help. Thanks.
Intern  Joined: 11 Jul 2013
Posts: 36
What is the remainder when a is divided by 4?  [#permalink]

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1
What is the remainder when a is divided by 4?
(1) a is the square of a odd integer.
(2)a is a multiple of 3.

Originally posted by domfrancondumas on 08 Aug 2013, 00:02.
Last edited by Zarrolou on 08 Aug 2013, 00:10, edited 1 time in total.
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Re: What is the remainder when a is divided by 4?  [#permalink]

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What is the remainder when a is divided by 4?

(1) a is the square of a odd integer.
Method #1:
a=1 reminder=1; a=9 reminder=1, a=25 reminder=1; a=49 reminder=1... I see a pattern, I am convinced that this is sufficient.
Method #2: $$a=(2k+1)^2$$ (where k is an integer)
$$a=4k^2+4k+1$$, $$a=4(k^2+k)+1$$ $$a$$ is a multiple of four plus one, hence the reminder will be one.
Sufficient

(2)a is a multiple of 3.
a=3 reminder=3, a=9 reminder=1.
Not sufficient.
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Re: What is the remainder when a is divided by 4?  [#permalink]

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ksear wrote:
What is the remainder when a is divided by 4?

(1) a is the square of an odd integer.
(2) a is a multiple of 3.

According to the book, the answer is A.

But according to me, the answer should be C since if we take the first statement and use the value 1 for a, it gives us 1 for the square of 1, what would be the remainder?
When we use 3 for a, the square of a will give us 9 which when divided by 4, gives us remainder of 1.

Thanks.
Kash.

1^2/4 = (4-3)^2/4 = (4^2 - 2.4.3 + 9)/4 = 4 + 6 + 9/4 = 10 + 9/4
so the remainder from 9/4 we have is 1 .
That's how we can realize why 1 appears as the remainder when 1^2 is divided by 4.

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What is the remainder when p is divided by 4  [#permalink]

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What is the remainder when p is divided by 4 ?
(1) p is the square of an odd integer.
(2) p is a multiple of 3.
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What is the remainder when p is divided by 4  [#permalink]

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shashankism wrote:
What is the remainder when p is divided by 4 ?
(1) p is the square of an odd integer.
(2) p is a multiple of 3.

(1) p is the square of an odd integer.

Lets try $$3^2, 7^2$$

$$3^2 = 9.$$
9 divided by 4 gives remainder 1.

$$7^2 = 49$$
49 divided by 4 gives remainder 1.

Any square of odd integer divided by 4 will give remainder 1. -------- (I is Sufficient)

(2) p is a multiple of 3.

12 is multiple of 4 hence will give remainder 0.
3 divided by 4 will give remainder 3.
6 divided by 4 will give remainder 2. etc...
Therefore multiple values. -------- (II is Not Sufficient)

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Re: What is the remainder when a is divided by 4?  [#permalink]

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shashankism wrote:
What is the remainder when p is divided by 4 ?
(1) p is the square of an odd integer.
(2) p is a multiple of 3.

Merging topics. Please refer to the discussion above.
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Re: What is the remainder when a is divided by 4?  [#permalink]

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ksear wrote:
What is the remainder when a is divided by 4?

(1) a is the square of an odd integer.
(2) a is a multiple of 3.

According to the book, the answer is A.

But according to me, the answer should be C since if we take the first statement and use the value 1 for a, it gives us 1 for the square of 1, what would be the remainder?
When we use 3 for a, the square of a will give us 9 which when divided by 4, gives us remainder of 1.

Thanks.
Kash.

(1) Let a = (2k+1)^2 as it is square of an odd integer = 4k^2 + 4k +1 . So when it is divided by 4, remainder =1. Sufficient
(2) a is a multiple of 3, i.e. 3,6,9,12,15,........ So when it is divided by 4, remainder = 3 , 2, 1, 0, So Not sufficient

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Re: What is the remainder when a is divided by 4?  [#permalink]

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