minimum number of parts?

Question

if n is a positive number , then (n^11-n) can be divided into the minimum number of parts by which among the following ?

(1) 6
(2)66
(3)33
(4)can not be determined


what it means by "minimum number of parts " in this problem ?

Answer 1

Basically it's asking for the largest of the given numbers that we know will be a factor of n^11-n. (E.g. 6 divides it into (n^11-n)/6 parts - if 6 is a factor - so the smallest number of parts comes from the largest factor.)

- From Fermat's little theorm we know n^11 ≡ n (mod 11), i.e. 11 | n^11 - n.
- Also from Fermat's little theorem, n^11 = (n^3)^3.n^2 ≡ n^3.n^2 ≡ n.n^2 ≡ n (mod 3), so 3 | n^11 - n.
- Finally, we know 2 | n^11 - n (n^11 is odd iff n is odd).

So we know 2, 3 and 11 will all be factors of n^11 - n, and since they are all coprime, we know 2×3×11 = 66 will be a factor of n^11 - n. So the answer is (1).