Question:
For some integer m, every even integer is of the form
(a) m
(b) m +1
(c) 2m +1
(d) 2m
Solution:
(c) We know that, even integers are 2, 4, 6,…
So, it can be written in the form of 2m
where, $m=$ Integer $=Z \quad[$ since, integer is represented by $Z]$
or $m=\cdots,-1,0,1,2,3, \cdots$
$\therefore \quad 2 m=\cdots,-2,0,2,4,6 \ldots$
Alternate Method
Let ' $a$ ' be a positive integer. On dividing ' $a$ ' by 2 , let $m$ be the quotient and f be the remainder. Then, by Euclid's division algorithm, we have
$\mathrm{a}=2 m+r$, where $\mathrm{a} \leq r<2 i . e ., r=0$ and $r=1$
$\Rightarrow \quad a=2 m$ or $a=2 m+1$
when, $a=2 m$ for some integer $m$, then clearly a is even.