Question

The product of two consecutive positive integers is divisible by $2.$Is this statement true or false? Give reasons.

• A. True
• B. False

Answer 1

The correct option is A True

Yes, the statement $"$the product of two consecutive positive integers is divisible by $2"$ is true.

Justification:

Let us assume the two consecutive positive integers $=n,n+1$

According to Euclid’s division lemma,

We know that,

$a=bq+r,$ where $0\le r<b$

For $b=2,$

we know that

$a=2q+r,$ where $0\le r<2\dots \left(i\right)$

Substituting $r=0$in equation$\left(i\right),$

We get,

$n=2q,$ is divisible by $2.$

$n+1=2q+1,$ is not divisible by $2.$

Substituting $r=1$ in equation (i),

We get,

$n=2q+1,$ is not divisible by $2.$

$n+1=2q+1+1$

$=2q+2,$ is divisible by $2.$

Now, we observe that, for$0\le r<2,$ one out of every two consecutive integers is divisible by $2.$

So, the product of the two consecutive positive numbers will also be even.

Hence, the given statement of product of two consecutive positive integers is divisible by $2.$ is true