Student: How do time and modular arithmetic relate to each other?

Mentor: Modular arithmetic deals with repetitive cycles of numbers and remainders. The time you use everyday is a cycle of 12 hours, divided up into a cycle of 60 minutes

Student: And those are divided into cycles of 60 seconds.

Mentor: Correct. Let's talk about the 12 hour cycle. Every time you pass 12 you start over with 1 again. This is mod 12 arithmetic. Let's set up a picture.

You put 0 through 11 in a circle. Then to figure out what the answer to a modular math question, you begin at zero and count around the circle (mmmm--) times. The number you end up on is the answer. Why don't you try 16 mod 12 on the circle I have provided.

Student: It would be 4. 16 mod 12 equals 4. I get it, but why not number around the circle from 1 to 12?

Mentor: First, your answer is right! Second, the reason we label the circle from 0 around to 11 will become clear in a minute. Let's try another. Now, what is 155 mod 12?

Student: That will take awhile.

Mentor: Are there any easier ways to do this? Do you notice any patterns?

Student: I don't think that I am following you.

Mentor: This is how it is tied to what we already know about remainders, and why we number from 0 to 11 instead of from 1 to 12. Look at what you did above. Isn't 16 mod 12 the same as the remainder when you divide 16 by 12?

Student: So, 16/12 = 1 remainder 4. The answer is four.

Mentor: Yes! So, we really want to label around the circle with every possible number we can get as a remainder when we divide by 12. That would be everything between 0 and 11 inclusive! Now, what is 155 mod 12?

Student: 155 divided by 12 is 12 with 11 left over. 155 mod 12 equals 11.

Mentor: Correct! You have the idea now.

Student: That seems easy.

Mentor: Now tell me how things would change if you used seconds?

Student: The modulus would change to 60 and it would be the same with minutes.

Mentor: Let's practice one like this. What is 254 mod 60?

Student: That would be 14 since 254/60 = 4 remainder 14.

Mentor: Good. You would deal with different sorts of time the same way. Military time is on a 24 hour cycle and that would mod 24. There are 12 months in a year so that one would also be mod 12. Just remember that modular arithmetic can be done with any finite repeating cycle. Can you think of any others?