Question

if n is an odd integer, then use euclid's division lemma to show that n2-1 is divisible by 8

Answer 1

n² - 1 is divisible by 8 if n is an odd integer

Step-by-step explanation:

n is an odd  number

hence n = 2k + 1

n² -  1= (2k + 1)² - 1

= 4k²  + 1  + 4k - 1

= 4k² + 4k

=  4k(k + 1)

two possible cases

k is odd or k is even

let say k  is odd

then k = 2a - 1   , k + 1 = 2a

= 4(2a - 1)2a

= 8a(2a - 1)

Divisible by 8

if k is even

k = 2a , k = 2a + 1

= 4(2a)(2a + 1)

= 8a(2a + 1)

Divisible by 8

Hence n² - 1 is divisible by 8 if n is an odd integer

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