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Puzzles of the Month + Solutions


December
2007January 2008, Puzzle nr 115 
Back to PuzzleoftheMonth
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!

© 2004 G.
Sarcone, www.archimedeslab.org
You can reuse content from Archimedes’ Lab
on the ONLY condition that you provide credit to the
authors (© G.
Sarcone and/or M.J.
Waeber) and a link back to our site. You CANNOT
reproduce the content of this page for commercial
purposes.

You're
encouraged to expand and/or improve this article. Send
your comments, feedback or suggestions to Gianni
A. Sarcone. Thanks! 

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.


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