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## A Mod Calendar for the New Year

Given this is the start of a new year, the following problem seems appropriate...

We assign a number N to months: 1 = January, 2 = February, ..., 12 = December. Write the twelve months of the year in the order offered by (5N+2)(mod 12), n = 1, 2, 3, ..., 12. What do you notice about the characteristics of the first seven months, of the next four months, and of the last month?

Side Problem: Investigate the relationship (5N+2)(mod 12), and its effect on the number set {1, 2, ..., 12}. That is, the relationship returns the set {1, 2, ..., 12}. Will this work for (5N+2)(mod x) for the number set {1, 2,...,x} for any positive integer x? Can you generalize for the expression (aN+b)(mod x), so that the relationship does work?

Hint: First, do the requested mod computations. Remember that 12(mod 12) = 0 = 12.

Now, study the new orderings of the months...

Solution Commentary: If the computations are done correctly, the new order of the months is July, December, May, October, March, August, January, June, November, April, September, and February.

That is, the first seven months have 31 days, the next four months have 30 days, and the last month has 28 (or 29) days.

Neat but unusual trivia, right!