Let m be a non zero integer and n be a positive integer Let R be the remainder obtained on dividing ...........

Quantitative Aptitude (CDS) Number System

Let m be a non-zero integer and n be a positive integer. Let R be the remainder obtained on dividing the polynomial xⁿ + mⁿby (x − m). Then,

R is a non-zero even integer

R is odd, if m is odd

R = s²for some integer s, if n is even

R = t³ for some integer t, if 3 divides n

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Explanation:

If R is the remainder obtained by dividing the polynomial xⁿ+mⁿ by (x-m), then (xⁿ+mⁿ-R) is divisible by (x-m).
Let f(x) =xⁿ+mⁿ-R
$\because$ f (x) is divisible by (x − m).
$\therefore$ f(m)=0
=>(m)ⁿ+(m)ⁿ-R=0=>R=2(m)ⁿ
As, m is a non-zero integer and n is a positive integer, then R is a non-zero even integer.