Question

6. Show that if a, b, m, and n are integers such that m ) 0, n ) 0, n i m, and a = b (mod rn ), then a = b (mod n)

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Answer:

It's much easier to give a direct proof. Suppose n,m,a,b are as in your hypothesis. a≡b(modx) is equivalent to x∣(b−a). Thus, m∣(b−a), so there is k such that km=b−a. Because n∣m, there is l such that ln=m. So kln=b−a, which is to say, n∣(b−a), or in other words a≡b(modn).

More briefly: By hypothesis, n∣m and m∣(b−a). The "divides" relation is transitive, so n∣(b−a) i.e. a≡b(modn).