1. Quote of the month: "Friendship is like money - easier made than kept" - Samuel Butler
2. New quiz AQQ300 about a divisibility by 6 puzzle
Good luck and try to be among the first ones!
1. Quote of the month: "Friendship is like money - easier made than kept" - Samuel Butler
2. New quiz AQQ300 about a divisibility by 6 puzzle
Good luck and try to be among the first ones!
1
That's a great observation! Here is my proof:
n⁵ - n is divisible by 5, because:
We have n⁵ - n = n(n - 1)(n + 1)(n² + 1).
We simply need to consider the general form of n: n = 10k ± a, where a = 0, 1, 2, …, 5 and k = 0, 1, 2, …, and evaluate the different possible cases for a:
If a = 5 or if a = 0, then n is a multiple of 5, so n⁵ - n is divisible by 5 since its expansion contains the factor n.
If a = 4, then either n + 1 (if n = 10k + 4) or n - 1 (if n = 10k - 4) is divisible by 5, so n⁵ - n is divisible by 5 since its expansion contains (n - 1). (n+1)
If a = 3, the units digit of n is 3, so n² + 1 ends in 0 and is therefore a multiple of 10, hence n⁵ - n is divisible by 5 since its expansion contains a factor of 10
If a = 2, n² must end in 4, and n² + 1 is a multiple of 5 since it ends in 5, which implies that n⁵ - n is divisible by 5
If a = 1, then either n – 1 (if n = 10k + 1) or n + 1 (if n = 10k – 1) is divisible by 5, so n⁵ – n is divisible by 5 since its expansion contains (n – 1)(n + 1).
