a^{ p − 1} ≡ 1 (mod p).Take advantage of Fermat's Little Theorem to compute the value 9^{5282} % 19 by noticing that 9^{18} ≡ 1 (mod 19). Show your work.
x ≡ 5 (mod 7)Note that 7 ⋅ 13 ⋅ 16 = 1456 and recall that the notation
x ≡ 8 (mod 13)
x ≡ 11 (mod 16)
a ≡ b (mod n)means that a % n = b % n, where % is the remainder operator. Show your work. (See notes on the Chinese Remainder Theorem.)
Additional Note: You need to pick an e such that gcd(e,φ(n)) = 1. The usual recommendation is to pick a prime number. However, you must still check that gcd(e,φ(n)) = 1 when e is prime, because it is possible that e divides φ(n). For example, in this problem picking e = 3 doesn't work because φ(n) = 1473168 is divisible by 3 and so gcd(3,1473168) = 3. There are no inverses of 3 modulo 1473168.
Another Additional Note: You are not allowed to pick e = 1.
Show your work and compute the values of E and D. Since M^{e} and E^{d} are very large numbers, you should use repeated squaring to compute these values. You may use a computer program or a spreadsheet to help you calculate these values, but you must write down all the intermediate steps from repeated squaring and show that E and D can be computed without using any numbers bigger than n^{2} = 2170223956224.