Happy birthday to me (part 1)

Number Theory Level 4

Let \(M/D\) be the date of my birthday, where \(M\) and \(D\) are the month and the day of the date, respectively. Then, there are some properties that relate \(M\) and \(D\). They are:

  • \(D\) is divisible by \(M\);
  • \(M\) represents the number of, not necessarily distinct, prime factors of \(D\);
  • If \(01/01^{st}\) is a thursday, then \(M/D\) is a friday.

After applying these properties, you have achieved a few possible dates. These possible dates can be grouped in sets, in function of every possible month, as follows:

\(S({M_i})=\{{D_1}, {D_2}, \ ..., \ {D_n}\}\).

Then, the last property would be:

  • The sum of possible days in a possible month is the smallest as possible.

Find \(M+D\).

Details and assumptions

  • As examples, the number of prime factors of \(8\) is \(3\), because \(8=2\cdot{2}\cdot{2}\), and the number of prime factors of \(20\) is \(3\), because \(20=2\cdot{2}\cdot{5}\)
  • Assume you are using Gregorian calendar in a non-bissextile year.

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