If the product of all positive divisors of n is n^2. Prove v(n) [the number of positive divisors function]=4
If n is congruent to k mod 40. Prove e^n is congruent to 2^k mod 55
Prove n^11 is congruent to n mod 33.
Thanks for any help!
If the product of all positive divisors of n is n^2. Prove v(n) [the number of positive divisors function]=4
If n is congruent to k mod 40. Prove e^n is congruent to 2^k mod 55
Prove n^11 is congruent to n mod 33.
Thanks for any help!
1) See result (2) here
2) What is that e? by the way What is $\displaystyle \phi(55)$?
3) Note that $\displaystyle 33=11\cdot{3}$ and that by Fermat's Little Theorem: $\displaystyle n^p\equiv{n}(\bmod.p)$ when p is prime (for all integers $\displaystyle n$)
So $\displaystyle n^{11}\equiv{n}(\bmod.11)$ (a)
And $\displaystyle n^{11}=(n^3)^3\cdot{n^2}\equiv{n^5}(\bmod.3)$ but $\displaystyle n^{3}\equiv{n}(\bmod.3)$ we have $\displaystyle n^5=n^3\cdot{n^2}\equiv{n^3}\equiv{n}(\bmod.3)$ thus $\displaystyle n^{11}\equiv{n}(\bmod.3)$ (b)
By (a) $\displaystyle 11|(n^{11}-n)$ and by (b): $\displaystyle 3|(n^{11}-n)$ thus $\displaystyle 33|(n^{11}-n)$ which shows that $\displaystyle n^{11}\equiv{n}(\bmod.33)$