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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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