Simplify the following?

Question

A) (-133 mod 23 + 270 mod 23) mod 23

B ) (177 mod 31 x 270 mod 31) mod 31

C) ( 89^3 mod 79)^4 mod 26

Answer 1

There is a problem... which operator has higher

precedence: the mod or the addition? In many systems,

the mod operator has equal precedence to multiplication

and division but higher precedence than addition

and subtraction because it is a division operation.

(a) mod calculated before addition....

-133 / 23 = -6 r 5 so -133 mod 23 = 5

270/23 = 11 r 17 so 270 mod 23 = 17

5+17=22

22 mod 23 = 22 <--- final answer

(b) all have equal precedence, so done left to right...

177 mod 31 = 22

22 * 270 = 5940

5940 mod 31 = 19

finally 19 mod 31 = 19

(c) exponents have higher proirity than mod per PEMDAS:

89^3 = 704969

704969 mod 79 = 52

52^4 = 7311616

7311616 mod 26 = 281216 with no remainder!

so the final answer is ZER0!