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Thread: euler phi function and others

  1. #1
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    euler phi function and others

    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!
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  2. #2
    Super Member PaulRS's Avatar
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    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 $\displaystyle n^{11}\equiv{n}(\bmod.33)$
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  3. #3
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    For the second problem, sorry it is supposed to be 2^n not e^n. And phi of 55 is 55(4/5)(10/11)= 40.

    Thanks for the help!
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