what is the highest power of 2 that can divide n12 -n8-n4+1 if n is odd?
Answers
Given : n¹² - n⁸ - n⁴ + 1 , n is odd integer
To find : Highest power of 2 that can divide
Solution:
n¹² - n⁸ - n⁴ + 1
= n⁸(n⁴ - 1) - (n⁴ - 1)
= (n⁴ - 1)(n⁸ - 1)
= (n⁴ - 1)(n⁴ + 1)(n⁴ - 1)
= (n⁴ - 1)²(n⁴ + 1)
= (n² + 1)²(n² - 1)²(n⁴ + 1)
as n is odd hence n² + 1 is divisible by 2
n² - 1 is divisible by 8
n⁴ + 1 is divisible by 2
Hence Divisible by
= (2)²(8)²(2)
= 2² * 2⁶ * 2
= 2⁹
9 is the power of 2 that can divide n¹² - n⁸ - n⁴ + 1
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