1.

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

Answer»

Solution:
True.
Justification:
Let a, a + 1 be two consecutive positive integers.
By 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)
Putting r = 0 in (i), we get
a = 2q, which is divisible by 2.
a + 1 = 2q + 1, which is not divisible by 2.
Putting r = 1 in (i), we get
a = 2q + 1, which is not divisible by 2.
a + 1 = 2q + 2, which is divisible by 2.
Thus for 0 ≤ r < 2, one out of every two consecutive integers is divisible by 2.
Hence, The product of two consecutive positive integers is divisible by 2.


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