If n is an odd integer,then show that n2-1 is divisible by 8
Answers
Answer:
Step-by-step explanation:
Hi my friend,
Let n be an odd number.
As we know all even numbers are in the form of 2k (Where k ∈ N)
Henceforth we also know all odd numbers are in the form of 2k + 1 (Where k ∈ W)
Therefore given to prove n² - 1 is divisible by 8
We know n = 2k + 1
Substitute n value in the given question.
=> (2k + 1)² - 1
=> (4k² + 4k + 1) - 1
=> 4k² + 4k + 1 - 1
=> 4k² + 4k + 0
=> 4k² + 4k
=> 4(k² + k)
Hence it can be divisible by 4. Since 4 is a factor.
Now take 4k² + 4k. It doesn't has any +1 or -1 attached to it. So it is the form of 2n.
Hence This term "4k² + 4k" is even. Hence it is divisible by 2 (Since remainder is 0)
Hence this term "4k² + 4k" is divisible by both 2 and 4.
Hence it must be divisible by their LCM.
LCM of 2 and 4 Is 8.
Hence "4k² + 4k" i.e. n² - 1 (Where n is an odd integer) is divisible by 8.
Hence Proved
Harith
Maths Aryabhatta
