SOLUTION: If n is a whole number then the largest number that n(n+1)(2n+1) is divisible by for all n is a) 2 b) 6 c) 10 d) 3 e) none

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Question 233402: If n is a whole number then the largest number that n(n+1)(2n+1) is divisible by for all n is
a) 2 b) 6 c) 10 d) 3 e) none

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Assuming n must be an integer (or a whole number or a natural number), here the keys to this problem:
  • All even numbers are divisible by two.
  • Whenever integers are multiplied, if one or more of the numbers is even, then product will be even.
  • If n is even, n+1 will be odd. Or if n is odd, then n+1 will be even. The point is that either n or n+1 must be an even number.
  • So n(n+1)(2n+1) must be even.
  • So n(n+1)(2n+1) must be divisible by 2.

The answer is definitely not (e). I cannot find any way to show that n(n+1)(2n+1) is always divisible by anything but 2 so I believe the answer is (a).