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Felix's Divisibility Troubles - DMS Discussion

Felix's Divisibility Troubles - DMS Discussion

Now that the Daily Magic Spells are back, we will be choosing one problem each week and talk a little about it. Make sure to join the discussion, let us know what you tried when you attempted the problem and come back to this forum every Wednesday for the chosen problem of the week!

From the week of September 18 to September 22 we chose Wednesday's problem (that you can find here):

Felix has a list of $12$ consecutive positive integers. In how many ways can he select $3$ of the numbers so that the sum of them is divisible by $3$?

26% of the students who attempted the problem got it right on their first attempt.

Probably one thing that was confusing (or scary!) about this problem was that it did not give a list of numbers, but rather said "Felix has a list of $12$ consecutive positive integers". Remember consecutive numbers come one after another, so for this problem, since Felix wants positive integers, the list of numbers can start with any number bigger than $0$ and increase by $1$ every time.

The problem wants us to count the number of ways Felix can choose $3$ numbers whose sum is divisible by $3$. This means the sum should be a multiple of $3$, or leave $0$ remainder when divided by $3$.

Remainders are one of the most important concepts in Number Theory, and you can do a lot with remainders! You already know more about them than you think: when we divide a number by $2$ we get either a reminder of $0$ or $1$, which tells us if the number is even or odd. Further, if we add two even numbers or two odd numbers their sum will be even, but if we add one even and one odd, their sum will be odd. Try to do something similar for division by $3$. Can you find any patterns?

Now let's return to the $12$ consecutive positive integers. Look at a few examples. What can you say about the remainders? Once you see the pattern with the remainders, you're ready to count and solve the problem! We'll discuss counting methods in the future.

Share your thoughts and questions below!