Question

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

Options

• True

• False

Answer 1

This statement is True.

Explanation:

Let the two consecutive positive integers = a, a + 1

We have,

a = bq + r

Where 0 ≤ r < b

For b = 2, we have a = 2q + r

Where 0 ≤ r < 2  ......(i)

Substituting r = 0 in equation (i),

We get,

a = 2q, is divisible by 2.

a + 1 = 2q + 1, is not divisible by 2.

Substituting r = 1 in equation (i),

We get,

a = 2q + 1, is not divisible by 2.

a + 1 = 2q + 1 + 1 = 2q + 2, is divisible by 2.

Thus, we can conclude that, for 0 ≤ 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 statement “product of two consecutive positive integers is divisible by 2” is true.