Clocks and Modular Arithmetic

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.

Make the circle read 0 through 11, so there are 12 hours total. To figure out the answer to a modular math question (in this case mod 12), begin at zero and count the number specified in the problem. The number you land 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?