Number Theory Worksheet
Dr. Holmes
July 19, 2007
1. Chinese Remainder Theorem
Solve the system of equations
x≡12 mod 137
x≡42 mod 125
Give the smallest positive solution.
2. Exponentiation by repeated squaring
Compute the following powers in modular arithmetic using the method
of repeated squaring (don’t use Fermat’s little theorem even if it ap-
plies).
(a) 421024 mod 1000
(b) 7261 mod 143
(c) 71115 mod 11
3. Compute the following powers in modular arithmetic using the theorem
of Fermat.
(a) 71115 mod 11
(b) 225471 mod 31
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