**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

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