prove that the product of two consecutive positive integer is divisible by 2
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Prove that the product of two consecutive positive integers is divisible by 2.
Sol:
Let, (n – 1) and n be two consecutive positive integers
∴ Their product = n(n – 1)
= 표2− 표
We know that any positive integer is of the form 2q or 2q + 1, for some integer q
When n =2q, we have
표2− 표 = (2푟)2− 2푟
= 4푟2− 2푟
2푟(2푟 − 1)
Then 표2− 표 is divisible by 2.
When n = 2q + 1, we have
표2− 표 = (2푟 + 1)2− (2푟 + 1)
= 4푟2+ 4푟 + 1 − 2푟 − 1
= 4푟2+ 2푟
= 2푟(2푟 + 1)
Then 표2− 표 is divisible by 2.
Hence the product of two consecutive positive integers is divisible by 2.
Sol:
Let, (n – 1) and n be two consecutive positive integers
∴ Their product = n(n – 1)
= 표2− 표
We know that any positive integer is of the form 2q or 2q + 1, for some integer q
When n =2q, we have
표2− 표 = (2푟)2− 2푟
= 4푟2− 2푟
2푟(2푟 − 1)
Then 표2− 표 is divisible by 2.
When n = 2q + 1, we have
표2− 표 = (2푟 + 1)2− (2푟 + 1)
= 4푟2+ 4푟 + 1 − 2푟 − 1
= 4푟2+ 2푟
= 2푟(2푟 + 1)
Then 표2− 표 is divisible by 2.
Hence the product of two consecutive positive integers is divisible by 2.
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Step-by-step explanation:
the product of two consecutive no. is divided by 2
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