A certain performer tries to impress his audience by guessing the birthday date of a volunteer (previously unknown to him).
The volunteer is requested to multiply the numerical value of the month of his birth by 31, to multiply the numerical value of the day by 12, to add the two products and announce the result, say N.

Upon getting the result the performer, considerably quickly, deduces the MM/DD of the relevant birthdate.

a. Devise a way to quickly solve 12*d+31*m=N.

b. Show that there is a unique solution for any N, evaluated as described above.

Assume 12d+31m = N = 12e+31p

Then 12(d-e) = 31(p-m).

12 and 31 are relatively prime, so 31 must divide (d-e).

But (d-e) is between -30 and 30, so it must be 0.

Therefore, we cannot have two different birthdays that result in the same N.