Hello I can't quite seem how to finish this problem:
The number 3750 satisfies f(3750) = 1000. Find a number a that has the following three properties:
(i) a º 7^3003 (mod 3750).
(ii) 1 < a < 5000.
(iii) a is not divisible by 7.
THANKS!!!
Hello I can't quite seem how to finish this problem:
The number 3750 satisfies f(3750) = 1000. Find a number a that has the following three properties:
(i) a º 7^3003 (mod 3750).
(ii) 1 < a < 5000.
(iii) a is not divisible by 7.
THANKS!!!
Note that $\displaystyle 7^{3003} \equiv 343 \equiv 7^3 \pmod{3750}$ (it can be shown quite trivially using Euler's Generalization, with $\displaystyle \varphi{(3750)} = 1000$).
Such a number does not exist, since if $\displaystyle a \equiv 7^{3003} \equiv 7^3 \pmod{3750}$, then $\displaystyle 7 | a$, and we have a contradiction.