Question

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

Answer 1

Answer:

yes because in consecutive positive integers one is compulsory even so even multiples with other is always even

let numbers are

2x+1,2x+2

product is 4x^2+6x+2

2(2x^2+3x+1)

so it is divisible by 2

Answer 2

Justification:

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

According to Euclid’s division lemma,

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

We get,

a = 2q,

is divisible by 2.

a + 1 = 2q + 1,

is not divisible by 2.

(ii)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.