The question :-

Determine the remainder when 2^112 is divided by 7. Your solution must be well written and your conclusion clearly formulated. Explain reasoning using both text and mathematical symbols. Only calculations are not accepted. All theorems in the course literature may be used without proof.

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my answer >

2^112 % 7 = ?

We know that 2^12mod(7)

:- (2)^24mod(7)

:- (2)^31mod(7)

:- (2^3)^37 (1)^37mod(7)

then it becomes => 2^11 (1)mod(7)

Then multiply by 2 then => 2^112 (2)mod(7)

Which means that if you divide 2^112 by 7, you get 2 as a remainder.

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The feedback:-

Hi, the solution is on the right track, but the steps leading to the conclusion that 2^1122(mod7) don't seem quite right.

It is not true that 2^111(mod7). Can you see how you can use the congruences from the previous step to determine the remainder of 2^112 when divided by 7?

A tip is to look in the course literature's section on modulus calculation. Why does using modulo calculus work to determine the remainder?