Legendre's Formula
Legendre's Formula states that
where is a prime and
is the exponent of
in the prime factorization of
and
is the sum of the digits of
when written in base
.
Contents
Examples
Find the largest integer for which
divides
Solution 1
Using the first form of Legendre's Formula, substituting and
gives
which means that the largest integer
for which
divides
is
.
Solution 2
Using the second form of Legendre's Formula, substituting and
gives
The number
when expressed in Base-2 is
. This gives us
. Therefore,
which means that the largest integer
for which
divides
is
.
Proofs
Part 1
We use a counting argument.
We could say that is equal to the number of multiples of
less than
, or
. But the multiples of
are only counted once, when they should be counted twice. So we need to add
on. But this only counts the multiples of
twice, when we need to count them thrice. Therefore we must add a
on. We continue like this to get
. This makes sense, because the terms of this series tend to 0.
Part 2
Let the base representation of
be
where the
are digits in base
Then, the base
representation of
is
Note that the infinite sum of these numbers (which is
) is
Problems
Introductory
- How many zeros are at the end of the base-
representation of
?
Olympiad
- Let
be numbers of factors
of the number
(that is,
and
). Find the least
such that
. (Turkey TST 1990)
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