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a Math question?
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-- G. Sarcone
does the photo represent?
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
Puzzles of the Month + Solutions
2007-January 2008, Puzzle nr 115
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# 115 Italiano Français
basic math knowledge.
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’)
convention, month numbers are assigned as follows:
January = 1, February = 2,
March = 3, etc...
Probability that the equation 13d +
33m = 13d' + 33m' can
be satisfied when...
d and d' are
2 different days:
d, d' ,
and 1 ≤ d, d' ≤ 31
and m and m' are
2 different months:
m, m' ,
and 1 ≤ m, m' ≤ 12
can reduce the equation:
13d + 33m = 13d' + 33m'
to simple factors:
13(d - d') = 33(m' - m)
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.
for any value of m, m', we obtain:
-11 ≤ (m' - m) ≤ 11
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'.
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).
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
© 2004 G.
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Math Facts behind the puzzle
common multiple & greatest common factor
Minimo comune multiplo; Plus petit
commun multiple; Mínimo común múltiplo;
gemeinsames Vielfaches; Kleinste gemene veelvoud.
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),
and 15 (=3x5)
is 120 (=2x2x2x3x5).
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
common factor (GCF), sometimes known as
the greatest common divisor, is useful for reducing
fractions to be in lowest terms. For example, 42/56
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.
puzzles of the month...
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