Question

The difference between the squares of two consecutive odd integers is always divisible by ?

A) 8 B) 2 C) 6 D) 4

Answer 1

$\sf\red{\underline{\underline{Answer:}}}$

$\sf{A) \ 8}$

$\sf\pink{To \ find:}$

$\sf{\implies{The \ difference \ between \ two \ consecutive}}$

$\sf{odd \ integers \ is \ always \ divisible \ by?}$

$\sf{A) \ 8 \ B) \ 2 \ C) \ 6 \ D) \ 4}$

$\sf\green{\underline{\underline{Solution:}}}$

$\sf{Let \ the \ any \ two \ consecutive \ odd}$

$\sf{integers \ be \ a \ and \ a+2}$

$\sf{(a+2)^{2}-a^{2}}$

$\sf{\implies{a^{2}+4a+4-a^{2}}}$

$\sf{\implies{4a+4}}$

$\sf{\implies{4(a+1)}}$

$\sf{\therefore{The \ difference \ between \ two}}$

$\sf{consecutive \ odd \ number \ is \ 4(a+1).}$

$\sf{Here, \ a \ is \ any \ odd \ integer.}$

$\sf{But, \ Odd \ number+1=Even \ number}$

$\sf{\therefore{4(a+1)=4(Any \ even \ integer)}}$

$\sf{But, \ 4\times(Any \ even \ integer) \ will}$

$\sf{be \ always \ divide \ by \ 8.}$

$\sf{\therefore{Difference \ between \ any \ two}}$

$\sf{consecutive \ odd \ numbers \ will \ be \ always}$

$\sf{divisible \ by \ 8.}$