Theorem (1): The product of any n consecutive integers is divisible by .
Proof (1):
For , and to show that the last expression is an integer it is sufficient to show that any prime p which occurs in
to at least as high as a power in
. Thus, we have to show that
is greater than or equal to
.
Note: The Symbol : If a is a fraction or an irrational number, the symbol
will be used to denote the integral part of a.
Now, , and the same is true if we replace p by
,
in succession, hence the result in question.
Theorem (2): If n is a prime, then is divisible by n.
For by the preceding is divisible by
and since n is a prime and r is supposed to be less than n,
is prime to n. Hence,
is a divisor of
and is divisible by n.
Thus, if a is a prime, all the coefficients in the expansion of except the first and last are divisible by n.
Homework:
1) Find the highest power of 5 contained in
2) If n is an odd prime, the integral part of is divisible by
.
More later,
Nalin Pithwa