The integer part of a number
The integer part of a real number x is denoted by [x] and it is by definition the largest integer that does not
exceed x. For example [1.3] = 1 and [−2.7] = −3. This means that it is the unique integer n that satisfies:
n≤x<n+1
From a practical point of view, this means that if we spot a number n that satisfies the above relation, then it
has to coincide with [x].
A fine application of this principle can be seen when we try to compute the largest integer exponent k to which
we have to raise a positive number a > 1, before it exceeds a given positive real number x. In other words, we
wish to find the largest integer number k for which:
ak ≤ x
If k is this exponent, then we necessarily have:
ak ≤ x < ak+1
If we take the logarithms of both sides, then we find:
k log a ≤ log x < (k + 1) log a
or
k≤
log x
<k+1
log a
This is precisely the kind of relation that we described above. Therefore:
log x
k=
log a
In particular the largest power of a prime number p that does not exceed x is:
log x
log p
1