The theory of numbers, by Robert D. Carmichael ...

ELEMENTARY PROPERTIES OF INTEGERS

27

tained in the denominator. This index for the denominator
is the sum of the expressions
a a a
LP I pP 3 [PL r 0 (B)
lP P2 IP 3I
The corresponding index for the numerator is
lI+[ - 2+l 1+ (C)
But, since n=a+B+... +X, it is evident that
pr > Lp [ + + + pr'
WpWpLp1'
From this and the expressions in (B) and (C) it follows that
the index of the highest power of any prime p in the numerator
of (A) is equal to or greater than the index of the highest power
of p contained in its denominator. The theorem now follows
at once.
COROLLARY. The product of n consecutive integers is divisible
by n!.
EXERCISES
i. Show that the highest power of 2 contained in 1ooo! is 2994; in Igoo! is 218"''
Show that the highest power of 7 contained in ioooo! is 71665G
2. Find the highest power of 72 contained in iooo!
3. Show that iooo! ends with 24) zeros.
4. Show that there is no number A such that 37 is the highest power of 3 contained in n!.
5. Find the smallest number n such that the highest power of 5 contained
in.! is 53. What other numbers have the same property?
6. If n=rs, r and s being positive integers, show that n! is divisible by (r!)';
by (s!)T; by the least common multiple of (r!)1 and (s!)r.
7. If n=at-[+pq+rs, where a, 3, p, q, r, s, are positive integers, then n! is
divisible by

a!o!!(q!)(s!).