Maharashtra BoardEnglish MediumSTD 10MathsReal Numbers2 Marks
Question
Prove that the product of two consecutive positive integers is divisible by $2$.
✓
Answer
Let, $( n -1)$ and n be two consecutive positive integers
$\therefore \text { Their product }=n(n-1)$
$=n^2-n$
We know that any positive integer is of the form $2 q$ or $2 q+1$, for some integer $q$ When $n =2 q$, we have
$n^2-n=(2 q)^2-2 q$
$=4 q^2-2 q$
$2(2 q-1)$
Then $n ^2- n$ is divisible by $2$ .
When $n=2 q+1$, we have
$n^2-n=(2 q+1)^2-(2 q+1)$
$=4 q^2+4 q+1-2 q-1$
$=4 q^2+2 q$
$=2(2 q+1)$
Then $n ^2- n$ is divisible by $2 $.
Hence the product of two consecutive positive integers is divisible by $2$ .
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