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