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"In our endeavour to understand reality we are somewhat like a man trying to understand the mechanism of a closed watch. He sees the face and the moving hands, even hears its ticking, but he has no way of opening the case. If he is ingenious he may form some picture of a mechanism which could be responsible for all the things he observes, but he may never be quite sure his picture is the only one which could explain his observations. He will never be able to compare his picture with the real mechanism and he cannot even imagine the possibility of the meaning of such a comparison".
A. Einstein & L. Infeld  # Previous Puzzles of the Month + Solutions

December 2007-January 2008, Puzzle nr 115 Back to Puzzle-of-the-Month page | Home Puzzle # 115  Italiano Français Difficulty level:  , basic math knowledge.

 Strange birthdate probabilities    What is the probability to find two people with two different birthdates, such that their respective birthday number multiplied by 13 added to their respective birth month number multiplied by 33 adds up to the same result? (Or said in different words, given 2 different days d and d’, and 2 different months m and m’, we should obtain: 13d + 33m = 13d’ + 33m’)    By convention, month numbers are assigned as follows: January = 1, February = 2, March = 3, etc...  To find: Probability that the equation 13d + 33m = 13d' + 33m' can be satisfied when... d and d' are 2 different days: d, d' , d d', and 1 ≤ d, d' ≤ 31 and m and m' are 2 different months: m, m' , m m', and 1 ≤ m, m' ≤ 12

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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More Math Facts behind the puzzle

Least common multiple & greatest common factor
Minimo comune multiplo; Plus petit commun multiple; Mínimo común múltiplo; K
leinstes 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.

 Results...
Common Factor & Common Multiplier Calculator
Enter 2 or 3 integers, then click the button to find their common factor or/and multiplier.
1st Number: 2nd Number: 3rd Number:
 Least Common Multiplier (LCM): Greatest Common Factor (GCF):

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