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

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

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New post 26 Apr 2016, 01:00
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If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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New post Updated on: 26 Apr 2016, 17:03
1
5
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.
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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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New post 26 Apr 2016, 06:10
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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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New post 26 Apr 2016, 08:43
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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If N = 775 × 778 × 781, what is the remainder when N is divided by 14?  [#permalink]

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New post 26 Apr 2016, 09:10
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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New post 26 Apr 2016, 09:13
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....

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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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New post 11 Jul 2017, 21:01
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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New post 12 Jul 2017, 03:42
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


Answer 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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New post 12 Jul 2017, 03:51
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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New post 22 Aug 2018, 06:42
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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New post 22 Aug 2018, 06:49
1
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

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Re: If N = 775 × 778 × 781, what is the remainder when N is divided by 14? &nbs [#permalink] 22 Aug 2018, 06:49
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