Question -
Answer -
Solution:
Yes, the statement тАЬthe product of twoconsecutive positive integers is divisible by 2тАЭ is true.
Justification:
Let the two consecutive positive integers = a,a + 1
According to EuclidтАЩs division lemma,
We have,
a = bq + r, where 0 тЙд r < b
For b = 2, we have a = 2q + r, where 0 тЙд r< 2 тАж (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 тЙд r < 2,one out of every two consecutive integers is divisible by 2. So, the product ofthe two consecutive positive numbers will also be even.
Hence, the statement тАЬproduct of twoconsecutive positive integers is divisible by 2тАЭ is true.