Q. 14.0( 13 Votes )

Prove that 16 divides n4 + 4n2 + 11, if n is an odd integer.

Answer :

Given here, n is an odd integer for some k Z

where Z is the set of all integers.


Since, we know that every odd integer is of the form 4k + 1 and 4k – 1.


Consider two cases:


Case 1: For n = 4k + 1






)


Therefore, it is divisible by 16.


Case 2: For n = 4k – 1






)


Therefore, it is divisible by 16.


Thus, n4 + 4n2 + 11 is divisible by 16.


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