# Proof Thery Exercise

[SOLVED] Proof Thery Exercise

1. Homework Statement

Prove the following statement, where m and n are integers:

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

2. Homework Equations

3. The Attempt at a Solution

X = 5 m + 6, Y = 5 n + 6
X * Y = (5 m + 6) * (5 n + 6)
X * Y = (25 mn + 30 m + 30 n + 36)
5k + 6 = (25 mn + 30 m + 30 n + 36)
5k = (25 mn + 30 m + 30 n + 36) - 6
5k = (25 mn + 30 m + 30 n + 30)
k = (5mn + 6m + 6n + 6)
5k + 6 = 5 (5mn + 6m + 6n + 6) + 6

I am stuck, maybe its with another method like "Contradiction or Contrapositive" proof, any help will be appreciated.

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Go all the way back to your third step:

If you obtained xy=(25mn+30m+30n+36) for integer m and n, can you rewrite the righthand side in terms of 5k+6 for some integer k? Your last step essentially does this.

At step 4, you decided to assume what you wanted to be true and then proceeded from there without finding a contradiction so you got stuck.

tiny-tim
Homework Helper
X = 5 m + 6, Y = 5 n + 6
X * Y = (5 m + 6) * (5 n + 6)
X * Y = (25 mn + 30 m + 30 n + 36)
Hi cannibal! I think you're frightened of mathematical proofs, and you're expecting them to be much longer than they really are.

So when you got to line 3, which is really the end, you thought "it can't be this easy", and you just carried on going … "So X*Y = 5k + 6, where k is the integer 5mn + 6m + 6n + 6." Hi cannibal! I think you're frightened of mathematical proofs, and you're expecting them to be much longer than they really are.

So when you got to line 3, which is really the end, you thought "it can't be this easy", and you just carried on going … "So X*Y = 5k + 6, where k is the integer 5mn + 6m + 6n + 6."  