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Math Expert V
Joined: 02 Sep 2009
Posts: 58434
If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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14 00:00

Difficulty:   55% (hard)

Question Stats: 65% (01:55) correct 35% (02:16) wrong based on 309 sessions

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If N = 775 × 778 × 781, what is the remainder when N is divided by 14?

A. 6
B. 7
C. 8
D. 9
E. 10

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Intern  Joined: 03 Mar 2013
Posts: 37
If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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2
8
Each of the three numbers may be written as $$14p+a$$, where p is an integer and a is the remainder we’re looking for. They’d look like this:

$$775 = (14p + a)$$
$$778 = (14q + b)$$
$$781 = (14r + c)$$

Example #1:
Now, lets multiple two of them and identify the remainder of that product.

$$775 * 778$$
$$= (14p + a)(14q + b)$$
$$= 14^2pq + 14qa + 14pb + ab$$
$$= 14 (14pq + qa + pb) + ab$$

All but one term are divisible by 14. So the remainder of that product (775*778) divided by 14 is the remainder of the product of their remainders (ab) divided by 14 only. Here, the remainder of 775/14 is 5 and the remainder of 778/14 is 8. Their product (ab) = 5*8 = 40. The remainder of 40/14 is 12. So the remainder of 775*778/14 is the same, 12.

Example #2:
Similarly,

$$775 * 778 * 781$$
$$= (14p + a)(14q +b)(14r + c)$$
$$= 14 (14^2pqr + 14pqc + 14pbr + 14aqr + pbc + aqc + abr) + abc$$

Again all but one term are divisible by 14. In other words, it's the remainder of that product of the numbers/14 is the remainder of abc/14, only.

General rule:
The remainder of the product of three large numbers divided by $$x$$ is the remainder of the product of their remainders divided by $$x$$.

Answer to the OP:
The following are easy to determine:
The remainder of 775/14 is 5
The remainder of 778/14 is 8
The remainder of 781/14 is 11

Finally, the remainder of (775 * 778 * 781)/14 is simply the remainder of the product of their remainders divided by 14:

$$abc / 14$$
$$= (5 * 8 * 11)/14$$
$$= 440/14$$
$$= 31 + 6/14$$

The remainder is thus 6, which corresponds to A.

Originally posted by InstantMBA on 26 Apr 2016, 04:55.
Last edited by InstantMBA on 26 Apr 2016, 17:03, edited 1 time in total.
General Discussion
Manager  B
Joined: 01 Jul 2010
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GMAT 1: 660 Q43 V38 GPA: 3.4
Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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3
775/14 leaves a remainder 5
778/14 leaves a remainder 8
781/14 leaves a remainder 11

5*8*11 =440
So the remainder will be the remainder of 440/14 which is 6

Ans A
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Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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raarun wrote:
775/14 leaves a remainder 5
778/14 leaves a remainder 8
781/14 leaves a remainder 11

5*8*11 =440
So the remainder will be the remainder of 440/14 which is 6

Ans A

IMHO

Same approach and same answer....
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Manager  B
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If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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Abhishek009 wrote:
raarun wrote:
775/14 leaves a remainder 5
778/14 leaves a remainder 8
781/14 leaves a remainder 11

5*8*11 =440
So the remainder will be the remainder of 440/14 which is 6

Ans A

IMHO

Same approach and same answer....

Sorry i just looked at the question and wrote how i would solve it.
I did not check how anyone else has solved it.
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Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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raarun wrote:
Sorry i just looked at the question and wrote how i would solve it.
I did not check how anyone else has solved it.

No issues bro , it doesn't matter whether a problem is solved , if you have something to share do so without any hesitation....

Abhishek
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Thanks and Regards

Abhishek....

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Manager  B
Joined: 24 Jul 2016
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Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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Bunuel wrote:
If N = 775 × 778 × 781, what is the remainder when N is divided by 14?

A. 6
B. 7
C. 8
D. 9
E. 10

if you divide each number by 14 then the remainder for the numbers are 5,8 & 11. i multiplied all the remainders and divided it by 14 (5x8x11/14). the remainder was 6.
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Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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Bunuel wrote:
If N = 775 × 778 × 781, what is the remainder when N is divided by 14?

A. 6
B. 7
C. 8
D. 9
E. 10

N = 775 x 778 x 781
Remainder of 775 x 778 x 781 /14 = {775 x 778 x 781 /14} = {5 x 8 x 11/14} = {5 x 4 /14} = 6

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Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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Write each no. as:

(14*55+5) (14*55+8) (14*55+ 11)

The first no. in the bracket is divisible by 14 and thus will leave no remainder.

The second nos. will leave a remainder.

We will get:
5*8*11= 440 which when divided by gives 6 as remainder.
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Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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Is it okay to apply the "last digit shortcut" here?

775*778*781

last digits multiplied together= 5*8*1= 40; here the last digit is 0

In order for the last digit of 14 to yield a 0, we have 6 as a remainder.
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Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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4
Bunuel wrote:
If N = 775 × 778 × 781, what is the remainder when N is divided by 14?

A. 6
B. 7
C. 8
D. 9
E. 10
20 seconds approach.

Ans A

Consider kudos if that helped.

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Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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shashankism wrote:
Bunuel wrote:
If N = 775 × 778 × 781, what is the remainder when N is divided by 14?

A. 6
B. 7
C. 8
D. 9
E. 10

N = 775 x 778 x 781
Remainder of 775 x 778 x 781 /14 = {775 x 778 x 781 /14} = {5 x 8 x 11/14} = {5 x 4 /14} = 6

Hello!

Two questions:

Why should we have to take 11 instead of 1?

How did you end with (5*4)/14?

Kind regards!
Senior Manager  S
Joined: 12 Sep 2017
Posts: 302
Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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Good night Bunuel !

Could you please provide the solution to this question?

Kind regards!
Intern  B
Joined: 19 Feb 2018
Posts: 3
Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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1
I found an easier approach to this sum. Please correct me if I am wrong.

775x778x781
First multiply 5x8 from the first two numbers. The units digits will be zero.

Then multiply this 0 with 1(Units digit of 781)

The units digit is 0.

Only remainder possible is 6 if the units digit is 0.
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Joined: 12 Sep 2015
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If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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Top Contributor
Bunuel wrote:
If N = 775 × 778 × 781, what is the remainder when N is divided by 14?

A. 6
B. 7
C. 8
D. 9
E. 10

First recognize that 770 is the biggest multiple of 14 that is less than 775, 778 and 781
770 = (14)(55), but let's just say that 770 = 14k (where k = 55)

This means 775 = 14k + 5, 778 = 14k + 8 and 781 = 14k + 11

So, 775 x 778 = (14k + 5)(14k + 8)
= 14²k² + (14k)(8) + (14k)(5) + 40
= 14²k² + (14k)(8) + (14k)(5) + 28 + 12
= 14²k² + (14k)(8) + (14k)(5) + 28 + 12
= 14(14k² + 8k + 5k + 2) + 12

Notice that 14(14k² + 8k + 5k + 2) is a multiple of 14.
So, we can let 14(14k² + 8k + 5k + 2) = 14q for some integer q
So, 775 x 778 = 14q + 12

This means 775 × 778 × 781 = (14q + 12) x 781
= (14q + 12) x (14k + 11)
= 14²kq + (14q)(11) + (14k)(12) + 132
= 14²kq + (14q)(11) + (14k)(12) + 126 + 6
= 14²kq + (14q)(11) + (14k)(12) + 126 + 6
= 14(14kq + 11q + 12k + 9) + 6
As we can see, 14(14kq + 11q + 12k + 9) is a multiple of 14
So, 14(14kq + 11q + 12k + 9) + 6, is 6 greater than a multiple of 14

So, if we divide 14(14kq + 11q + 12k + 9) + 6, the remainder will be 6

Cheers,
Brent

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