Previous Puzzles of the Month + Solutions

We can reduce the equation:
13d + 33m = 13d' + 33m'
to simple factors:
13(d - d') = 33(m' - m)

Since 13 and 33 are coprime numbers, then their least common multiple is 13 x 33. So, for the equation 13(d - d') = 33(m' - m) to be satisfied we must have at least (m' - m)mod13 = 0. Or said in other words, the difference (m' - m) should be a multiple of 13.

But, for any value of m, m', we obtain:
-11 ≤ (m' - m) ≤ 11
and for this range of values the equation is satisfied only for m' = m when d = d', but... it is given that months and dates are different: m m' and d d'.

Hence the equation is not satisfied for any values of d, d', m and m'. Therefore, the probability to find two people with two different birthdates, such that 13d + 33m = 13d' + 33m', is exactly 0 (zero).

The 5 Winners of the Puzzle of the Month are:
Amelia Smith, USA
Fabio Cirigliano, Italy
Barbara Matteuzzi, Italy
Rohan Pillai, India
Rupesh Kumar Navalakhe, India

Congratulations!

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© 2004 G. Sarcone, www.archimedes-lab.org
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More Math Facts behind the puzzle

Least common multiple & greatest common factor
Minimo comune multiplo; Plus petit commun multiple; Mínimo común múltiplo; Kleinstes gemeinsames Vielfaches; Kleinste gemene veelvoud.
Massimo comun divisore; Plus grand commun diviseur; Máximo común divisor; Größter gemeinsamer Teiler; Grootste gemene deler.

A common multiple is a number that is a multiple of two or more numbers. The common multiples of 3 and 4 are 0, 12, 24, ...etc.
The least common multiple (LCM) of two (or more) integers is the smallest number (not zero) that is a multiple of both. For instance, the least common multiple of 8 (=2x2x2), 12 (=2x2x3), and 15 (=3x5) is 120 (=2x2x2x3x5).

When adding or subtracting fractions, it is useful to find the least common multiple of the denominators, often called the lowest common denominator. In this sum 5/6 + 2/21, the lowest common denominator is 42. In fact, 5/6 + 2/21 = 35/42 + 4/42 = 39/42

The greatest common factor (GCF), sometimes known as the greatest common divisor, is useful for reducing fractions to be in lowest terms. For example, 42/56 = (3x14)/(4x14) = 3/4.

Here below are two interesting math tools that will help you find factors for any given number, or a common factor and multiplier for any couple of integers.