How do I Wear All Combinations of my Clothes at Least Once? I have 5 shorts and 13 t-shirts. My system is simple - I hang them sequentially on my rack and each day I pick up the front t-shirt and the front pair of shorts.
At the end of the day, I move each piece of clothing to the end of the line and the next day I pick up the "new" first in line.
I have the following questions:


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*Can I prove that I'm wearing every possible combination of shorts and t-shirts? (I can brute force this and the answer is, yes I am). Is the answer to simply ensure that the HCF of the number of t-shirts and shorts is 1?

*I'm moving to Canada soon and will have three layers of clothing. So I will then have an additional two sets of clothing - an inner layer, and an outer jacket. If I use the same "first in first out" system, will I still run through every combination of clothing as long as the HCF is 1?

*Is the branch of mathematics that deals with this combinatorics?
 A: Here is an answer from a group-theoretic perspective.
What you are essentially doing is looking at an action of the group $ G := C_{13} \times C_5 $ on the set consisting of all the pairs $ (\alpha,\beta) $ where $ \alpha $ is from the ordered set of t-shirts and $ \beta $ is from the ordered set of shorts. Rephrased in this way, the question is whether the element $ (g,h) $ of $ G $ (where $ g $ and $ h $ are generators of $ C_{13} $ and $ C_5 $ respectively) has an orbit of order $ 13 \cdot 5 $ (because this is the number of choices of clothing) on the set of clothing.
But notice that the set of clothing itself has a natural group structure isomorphic to $ G $ (namely, send $ (g^i,h^j) \leftrightarrow (\alpha_i,\beta_j) $ where $ \alpha_i $ is the $ i$th t-shirt and $ \beta_j $ is the $ j$th pair of shorts). Hence the action we are interested in is the group action of $ G $ on itself by left multiplication; and so since $ G $ is a product of cyclic groups of coprime order, the action of $ (g,h) $ on itself has order $ 13 \cdot 5 $ as conjectured.
So to answer your first question: yes, it is because $ \gcd(5,13) = 1 $; and your reasoning for question 2 is correct as well, as long as all three numbers are pairwise coprime.
To answer question three, although I phrased my answer in group-theoretic terms, it is indeed a question in combinatorics. Further, it is even a number theory question (as the comments pointed out) - most of the elementary theory of finite abelian groups is highly number-theoretic.
