Can you answer number 1? Also, if you have a link to a solution manual for this

Question

Can you answer number 1? Also, if you have a link to a solution manual for this book would you mind posting it. I can't seem to find one anywhere.

1. Le t n be a positive integer and let m be a positive divisor of n. If a and b are integers with a congruent b mod n, prove that a congruent mod m

Exercises I. Let n be a positive integer and let n be a positive divisor of n. If a and b are integers with a Io b mod n, prove that a b mod m. 2. For each nonnegative integer i, what is the least residue modulo 9 of 10?

Explanation / Answer

n is a positive integer and m is a divisor of n , i.e. m|n

again, ab (mod n) ==> n|(a-b) and m|n

so by transitivity of divisibility property (i.e. if, a|b and b|c the, a|c) we get, m|n and n|(a-b) then, m|(a-b)

i.e. ab (mod m)

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