Re: Practice with deductions

1 and 2. "If A then B" does NOT imply "if B then A".

in both A is "if I have my keys". That says nothing about "if I can open the door".

bi. An obvious way to prove that is to **do** the multiplications indicated on the right.

bii. You want to prove that n^4- n^2+ 1= 3k+ 1 for some number k. Subtracting 1 form both sides, n^4- n^2= 3k. Now look again at bi.

Re: Practice with deductions

Thanks Hallsofivy but I'm still got some doubts in how to work my way out in part b(ii)

Re: Practice with deductions

Proove first n(n-1)(n+1) is always divisible by 3 .It will allow you to use bi

Re: Practice with deductions

What Cartesius24 said! Think about the **three** numbers n-1, n, and n+ 1. ONE of those MUST be divisible by 3. Do you see why?

Re: Practice with deductions

Not really, I'm totally confused in how to see how this numbers can be divided by 3.

Re: Practice with deductions

Seriously? Don't you see that if you have three **consecutive** numbers one of them **must** be a multiple of 3?

For a formal proof, use the fact that any integer is of the form 3k or 3k+ 1 or 3k+ 2 for some integer k.

If n itself is not divisible by 3 then it must be of the form 3k+ 1 or 3k+ 2.

If n= 3k+ 1 then n- 1= 3k is divisible by 3. If n= 3k+ 2, then n+1= 3k+ 3= 3(k+ 1) is divisible by 3.