114 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

LEAST COMMON MULTIPLE

270. Th. If a positiv integer m is a common multiple of a
set of integers and every positiv common multiple of these integers
is @ multiple of m, then m 1is the least common multiple of these
integers.

Let n be any positiv common multiple of the given integers.

Then by the hypothesis n > m.

Hence n = gm, where ¢ is positiv.

That is, all the common multiples are in the series m, 2m,
3m, -

Now,if 1 < ¢, 1-m < g-m,orm < gm

Thus that one of the positiv common multiples for which
g = 1, namely m itself, is less than any positiv common mul-
tiple for which ¢ > 1, that is, less than any other positiv
common multiple.

Therfor m is the least common multiple.

271. Th. a—-=0~b-=0Da X b= |ab| : (a(XRDd)

Since neither a nor b is zero, |ab| is positiv.

Hence |ab| : (a X b) is positiv.

Moreover |ab| : (e D) = |a|(|b] : (a R b))

= (la] : (e X)) 5]

Therfor |ab| : (e ®b) is a positiv common multiple of
la| and |b|, and hence of @ and b.

Moreover every positiv common multiple of @ and b is a

multiple of |ab| : (¢ X b) §§ 269, 70.
Therfor |ab| : (e X b) is the least common multiple of a
and b. § 270.
Ex.1. 6X8=(6X8):(6R8) = (6X8) :2=24
Ex. 2. Find the least common multiple of 24 and 40.
Ex. 3. Find 212 X 316.
Ex. 4. Find 251 X 353.

272. Th. n>2>a~n>3>b>Dn>>a X by and conversely.
In words: Any common multiple of two integers is a multiple
of their least common multiple; and conversely, any multiple of