Proofs!! Prove the statement?

Question

Prove the statement where M and N are integers.

if x=5m+6 and y=5n+6, then xy=5k+6 for some integer k.

Answer 1

xy = (5m+6) (5n+6) = 25mn + 30m + 30n + 36 = 5(5mn + 6m + 6n + 6) + 6, so xy=5k+6 for k = (5mn + 6m + 6n + 6)

Answer 2

xy is equal to (5m+6)*(5n+6) = 25mn + 30(m+n)+36 using the product of binomials.

Let k = mn. Since m and n are integers, then k is also an integer by the closure property of multiplication
Also, choose k such that m+n=k

therefore, we write xy as xy=25k^2 + 30k + 36 which is now a perffect square trinomial, which we can factor as

xy = (5k+6)^2 . Note that since k is an integer, 5k+6 is also an integer (closure), and so is (5k+6)^2.

I think the problem should be xy=(5k+6)^2 for some k

Answer 3

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