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remainder question - mgmat

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remainder question - mgmat [#permalink] New post 10 Apr 2013, 17:02
hi, this question is from the number properties book, 4th edition, ch. 10 q. 23

apologies if this is in the wrong thread


you're given background info, integer x has a remainder of 5 when divided by 9, integer y has a remainder of 7 when divided by 9

question: what is the remainder when 5x - y is divided by 9?


i solved this by picking numbers, but i think that it's probably inefficient.

i did 2 examples and made sure i got the same number

taking info from above, x has remainder of 5 so:
x = 14, remainder is 5
5x = 70, remainder is 7

x = 23, remainder is 5
5x = 115, remainder is 7

i recall in chapter 10, it only mentions that if you multiply the remainders of x and y, it is equivalent to finding the remainder of xy, but i dont recall it saying that multiplying x by 5 is the same as multiplying Rx by 5 (remainder of x multiplied by 5)

could someone provide the explanation/proof/algebra of why this works?
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Re: remainder question - mgmat [#permalink] New post 10 Apr 2013, 19:36
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integer x has a remainder of 5 when divided by 9, integer y has a remainder of 7 when divided by 9

question: what is the remainder when 5x - y is divided by 9?

Let x = 9p+5 and y = 9q+7 where p and q are integers.

5x - y = 45p +25 - 9q - 7 = 45p - 9q + 18 = 9 ( 5p - q + 2 ) which is a multiple of 9.

So,the remainder will be zero.

I've explained more algebraically. Since we are dividing all the numbers by 9,we can easily find the remainder in the following way :

x gives remainder => 5
y gives remainder => 7.
5x-y gives remainder => 5 * 5 - 7 = 25 - 7 = 18.
18 when divided by 9,gives remainder 0.
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Re: remainder question - mgmat   [#permalink] 10 Apr 2013, 19:36
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