State

Euler’s Theorem concerning the congruence class of powers of an integer.

Determine whether or not 221 is a prime number and, if not, find its factors. Suppose that

your calculator is incapable of multiplying together two numbers the size of whose product

exceeds 500, but division is unaffected. Quoting any general result you require, show how to

calculate the positive remainder of 5^1158 when it is divided by 221 and give its value.

Ok i have shown that 221 isnt prime via prime factorisation:

13*17=221

From this i have used fermats theorem as both are prime such that the units of 13=12 and 17=16

As a result 5^12mod221=1mod221

Therefore 5^(12*96)*5^6=5^1156mod221

= 5^6 mod 221= 155 mod 221

I done the same with 17 and came out with the same thing, alternatively i used units of 221= 192 generated from:

221(1-1/17)(1-1/13) again via the same process got the 155mod221

Just wondering if my method is correct, or because of the condition, "your calculator is incapable of multiplying together two numbers the size of whose product exceeds 500" it is incorrect

Many thanks in advance.