Question

QUES.. ꜰᴏʀ ᴀɴʏ ᴘᴏꜱɪᴛɪᴠᴇ ɪɴᴛᴇɢᴇʀ ɴ, ᴘʀᴏᴠᴇ ᴛʜᴀᴛ ɴ3 – ɴ ɪꜱ ᴅɪᴠɪꜱɪʙʟᴇ ʙʏ 6.

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Answer 1

Answer:

n3 – n = n(n2 – 1) = n(n+1)(n – 1) = (n – 1)n(n+1) = product of three consecutive positive integers.

Now, we have to show that the product of three consecutive positive integers is divisible by 6.

We know that any positive integer n is of the form 3q, 3q + 1 or 3q + 2 for some positive integer q.

Now three consecutive positive integers are n, n + 1, n + 2.