Question

What is the ANSWER and REMAINDER of 2^63 divided by 5

Please show work and please also come up with a way to do it with a pattern of less than 10 digits. (try not to have it so complicated or long- needs to cover half a page)

Answer 1

Notice how 2^2 = 4 is one short of 5, so this means we can say 4/5 = 0 remainder 4 or we can write that "remainder 4" as "remainder -1" to say the same basic thing

Using modulus arithmetic, we can say...

2^2 = 4 = -1 (mod 5)

(2^2)^31 = (-1)^31 = -1 (mod 5)

2*(2^2)^31 = 2*(-1) (mod 5)

2^63 = -2 = 3 (mod 5)

-------------------------------------------

So the remainder of 2^63 divided by 5 is the value 3

Answer: 3