Prove that "The product of any three consecutive even natural numbers is divisible by 16 " - Maths - Playing with Numbers

Question

Prove that "The product of any three consecutive
even natural numbers is divisible by 16 ​"

Koka Sri Lakshmi Divya Sai · Accepted Answer

Dear Student

Let the three consecutive even natural numbers be 2n,2n+2 and 2n+4 where n∈NNow,2n(2n+2)(2n+4)=2n×2(n+1)×2n+2                               =8n(n+1)(n+2)Here there will two cases either n is even or oddCase I: When n is evenLet n=2mThen 2n(2n+2)(2n+4)=8n(n+1)(n+2)                               =8×2m2m+12m+2                                =16m2m+12m+2Clearly 16m2m+12m+2 is divisible by 16∴ 2n(2n+2)(2n+4) is divisible by 16Case II;When n is oddLet  n=2m-1Then2n(2n+2)(2n+4)=8n(n+1)(n+2)                               =8×2m-12m-1+12m-1+2                               =16m2m-12m+1Clearly  16m2m-12m+1 is divisible by 16  ∴ 2n(2n+2)(2n+4) is divisible by 16Hence the product of any three even natural  umbers is divisible by 16

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Prove that "The product of any three consecutive even natural numbers is divisible by 16. ​"

Prove that "The product of any three consecutive even natural numbers is divisible by 16. "