Solved: Prove Thery Exercise: 5m + 6, 5n + 6 = 5k + 6

[cannibal]

[SOLVED] Proof Thery Exercise

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.

Homework Equations

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.

 

[jhicks]

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]

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 …

All you had to do was add:

"So X*Y = 5k + 6, where k is the integer 5mn + 6m + 6n + 6."

 

[cannibal]

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 …

All you had to do was add:

"So X*Y = 5k + 6, where k is the integer 5mn + 6m + 6n + 6."

Oh i see, in i was really expecting the problem to be much longer, that is why i got stuck, Thank you very much for your help "tiny-tim" and "jhicks".