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# What is the remainder when 1044 * 1047 * 1050 * 1053 is

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What is the remainder when 1044 * 1047 * 1050 * 1053 is [#permalink]  23 Oct 2009, 14:04
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What is the remainder when 1044 * 1047 * 1050 * 1053 is divided by 33?

A. 3
B. 27
C. 30
D. 21
E. 18
[Reveal] Spoiler: OA

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Last edited by Bunuel on 09 Jul 2013, 09:42, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: Remainders [#permalink]  23 Oct 2009, 22:35
Opps! Really a hard one. I cannot figure out how to quickly solve it!! Please help someone.
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Re: Remainders [#permalink]  23 Oct 2009, 23:22
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Expert's post
This one really needs some calculations and I don't know how it can be done more quickly:

Let's find the closest multiple of 33 to these numbers: it's 1056.

(1056-12)(1056-9)(1056-6)(1056-3) every term after simplification will have 1056 as its multiple except the last one which will be 12*9*6*3.

So the remainder will be the same when 12*9*6*3 is divided by 33.

The same way here 12*6*3=216 remainder when divided by 33 =18,
9*18=162 remainder=30

Done

I believe there is a easier solution...
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Re: Remainders [#permalink]  24 Oct 2009, 06:35
I think it's easier: because all of the term divisible by 3 so the remainder of the multiply of them will be the remainder when after dividing by 3 then dividing by 11. This can only be 3 because all other choice is greater than 11
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Re: Remainders [#permalink]  24 Oct 2009, 08:22
Expert's post
ngoctraiden1905 wrote:
I think it's easier: because all of the term divisible by 3 so the remainder of the multiply of them will be the remainder when after dividing by 3 then dividing by 11. This can only be 3 because all other choice is greater than 11

That's not right:

Consider this 6*6*6 divided by 33. What is remainder?

According to you logic: as all of them are divisible by 3 remainder must be less than 11, but in this case remainder is 18>11.

For the original question the answer IS 30. But my point was that the solution I provided is not easy, so I wonder if there is some easier way to do the same.
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Re: Remainders [#permalink]  24 Oct 2009, 09:51
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To find the remainder, we need to first find what are the remainders when 1044, 1047, 1050 and 1053 are divided by 33 individually.

1044/33=31.......21
1047/33=31.......24
1050/33=31.......27
1053/33=31.......30

Now we need to find the remainders when 21*24 and 27*30 are divided by 33.

21*24/33=15......9
27*30/33=24......18

Finally we need to find the remainder when 9*18 is divided by 33:

9*18/33=4........30

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Re: Remainders [#permalink]  24 Oct 2009, 20:02
Economist wrote:
To find the remainder, we need to first find what are the remainders when 1044, 1047, 1050 and 1053 are divided by 33 individually.

1044/33=31.......21
1047/33=31.......24
1050/33=31.......27
1053/33=31.......30

Now we need to find the remainders when 21*24 and 27*30 are divided by 33.

21*24/33=15......9
27*30/33=24......18

Finally we need to find the remainder when 9*18 is divided by 33:

9*18/33=4........30

Thanks, this is indeed a easier way. It also seems to work for any number

Still a bit long solution......
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Re: Remainders [#permalink]  24 Oct 2009, 23:41
rlevochkin wrote:
OA is C.

Any OE? Coz answer C is apparent but how to solve is the key here.
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Re: Remainders [#permalink]  26 Oct 2009, 18:52
2
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Key to this remainder problem is the following properties,
you can multiply the remainders as long as you correct the excess remainders.

1044/33 yields R21
1047/33 yields R24 (Don't need to calculate as 1047 is 3 integers away from 1044)
1050/33 yields R27
1053/33 yields R30

Now, multiply remainders and correct the excess remainders.

R21* R24 * R27 * R30
Solve the problem till you get a remainder less than 33.

since 1044 and 33 are big numbers, it is possible to make arithmetic mistake here. Is there away to solve this problem
efficeintly?
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Re: Remainders [#permalink]  26 Oct 2009, 23:49
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calculate the remainder when 1044 is divided by 33 = 21

so the eqn becomes 21*24*27*30 / 33

= 27*27*21/33

considering negative remainders we get -6*-6*21/33

= 36*21/33

= 3*21/33

= 63/33

= 30.

I will go with option C
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Re: Remainders [#permalink]  26 Oct 2009, 23:57
deepakraam wrote:
calculate the remainder when 1044 is divided by 33 = 21

so the eqn becomes 21*24*27*30 / 33

= 27*27*21/33

considering negative remainders we get -6*-6*21/33

= 36*21/33

= 3*21/33

= 63/33

= 30.

I will go with option C

Wow!! Everyday I am learning new thing is this forum. Thanks.
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Re: Remainders [#permalink]  27 Oct 2009, 20:54
useful insight..
thanks for the good question.
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Re: Remainders [#permalink]  08 Nov 2009, 19:13
smtripathi wrote:
I guess, we can prove this by taking two numbers in general format :

N1=dQ1+R1 ......1
N2=dQ2+R2 ......2

so, if we multiply them to find the remainder, it will become :
N1*N2=d^2*Q1Q2+dQ2*R1+dQ1*R2+R1*R2
or, =d[d*Q1*Q2+Q2*R1+Q1*R2] + R1*R2
so, we have remainder R1*R2 , but this may be bigger than d, so we need to rationalize R1*R2 to get the actual remainder.

I tend to remember things if I prove them .. so thought of adding my two cents.

good one . this can be extended to the SUM as well.

thanks
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Re: Remainders [#permalink]  12 May 2011, 06:37
good method by Bunuel.
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Re: Remainders [#permalink]  22 Aug 2013, 11:24
Bunuel wrote:
This one really needs some calculations and I don't know how it can be done more quickly:

Let's find the closest multiple of 33 to these numbers: it's 1056.

(1056-12)(1056-9)(1056-6)(1056-3) every term after simplification will have 1056 as its multiple except the last one which will be 12*9*6*3.

So the remainder will be the same when 12*9*6*3 is divided by 33.

The same way here 12*6*3=216 remainder when divided by 33 =18,
9*18=162 remainder=30

Done

I believe there is a easier solution...

Different sort of PS..... rarely may see this. But solution by bunuel is superb...........
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Re: What is the remainder when 1044 * 1047 * 1050 * 1053 is [#permalink]  23 Aug 2013, 02:46
Expert's post
rlevochkin wrote:
What is the remainder when 1044 * 1047 * 1050 * 1053 is divided by 33?

A. 3
B. 27
C. 30
D. 21
E. 18

We can play with the questions the way we like to.... That's the beauty of remainder questions.

When 1044, 1047, 1050, and 1053 divided by 33 individually will give the remainders as 21, 24, 27, and 30 respectively.

Now as per rule, remainders are always non-negative, but still we can consider the negative remainders for the calculation as long as we convert them is positive remainder at the end.

So 21, 24, 27, and 30 when divided by 33 will give the negative remainders as -12, -9, -6, -3

(-12 * -9)(-6 * -3) --------> 108 * 18 ------->Positive Remainder -----> 9 * 18 --------> 162/33 ------Negative Remainder -----> -3 -------> Positive Remainder -----> -3 + 33 = 30 (Always add the divisor in to the negative remainder to obtain positive(correct) remainder)
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Re: What is the remainder when 1044 * 1047 * 1050 * 1053 is [#permalink]  23 Aug 2013, 08:31
Though I took a while to solve this but I was trying to recollect something which I learnt when I was in love with Remainder theorem.

(1044 x 1047 x 1050 x 1053)/33
(cancelling the common factor 3)
=>(1044 x 1047 x 1050 x 351)/11
Denoting remainder using []
[1044/11]= [10] is remainder. Similarly,
[1047/11]= [2]
[1050/11]= [5]
[351/11]= [10]
which gives: [(10 x 10 x 10)/11] = [10] as remainder.

Since we initially cancelled the common factor 3. The final remainder will be 10 x 3 =30 i.e. option C.
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Re: What is the remainder when 1044 * 1047 * 1050 * 1053 is [#permalink]  23 Aug 2013, 09:19
Expert's post
vabhs192003 wrote:
Though I took a while to solve this but I was trying to recollect something which I learnt when I was in love with Remainder theorem.

(1044 x 1047 x 1050 x 1053)/33
(cancelling the common factor 3)
=>(1044 x 1047 x 1050 x 351)/11
Denoting remainder using []
[1044/11]= [10] is remainder. Similarly,
[1047/11]= [2]
[1050/11]= [5]
[351/11]= [10]
which gives: [(10 x 10 x 10)/11] = [10] as remainder.

Since we initially cancelled the common factor 3. The final remainder will be 10 x 3 =30 i.e. option C.

Yeah, This is also a fantastic method and, if I remember correctly, has been discussed in Mr. Arun Sharma's CAT QA book
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Re: What is the remainder when 1044 * 1047 * 1050 * 1053 is   [#permalink] 23 Aug 2013, 09:19
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