Struggling with modular arithmetic problems.

Question

I've been struggling with this modular arithmetic problem for a while, I could solve the simpler problems where we have to find out what day it is or what hour is is after a certain amount of time has passed but I can not wrap my head around this problem from my past quiz, where do we start, how do we go around and solve this? Any sort of help is appreciated.

Question Assume that you are given the task to mark maths exams. Every script takes you 7 minutes to mark. You need to mark $ 56 ^{27} $ scripts. If you start at midnight, at what minute past the hour will you be done? For example, if you finish five days later at 11:23 am, then this counts as 23 minutes past the hour.

Answer 1

you need to mark $56^{27}$ scripts and each one of these takes you $7$ minutes so the total time should be $56^{27}\times7$ minutes. Since we do not care what hour we finish (finishing at $11:57$ and $03:57$ gives the exact same result). So it is obvious we need to work $\mod 60$. In other words we need to evaluate $$56^{27} \cdot7 \mod 60$$

Remember that $(a\cdot b) \mod c= ((a \mod c)\cdot(b \mod c)) \mod c$ . Therefore we get

$$((56^{27} \mod 60)\cdot(7 \mod 60))\mod 60\\(((56^2)^{13}\cdot56 \mod 60)\cdot7)\mod 60$$

Apply the same rule again and get :

$$(((((56^2 )^{13}\mod 60\cdot (56 \mod 60))\mod 60)\cdot 7)\mod 60$$

Also remember: $(a^b)\mod c=((a\mod c)^b)\mod c$. Here we get

$$((((((56^2\mod60)^{13}\mod60)\cdot56)\mod60)\cdot7)\mod60\\(((((3136\mod60)^{13})\mod60)\cdot56)\mod60)\cdot7)\mod60\\((((16^{13}\mod60)\cdot56)\mod60)\cdot7)\mod60$$ We can write $16^{13}=(16^2)^6\cdot 16$ and apply the two modulo rules , the product and exponent one respectively and get:

$$((((((16^2\cdot16\mod60)\cdot(16\mod60))\mod60)\cdot16)\mod60)\cdot56)\mod60)\cdot7)\mod60$$ These are really easy to compute and get to :

$$((16\cdot56\mod60)\cdot7)\mod60\\((896\mod60)\cdot7)\mod60\\392\mod60\\=32$$