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

ELEMENTARY PROPERTIES OF INTEGERS

21

numbers am, = adu, where a is any integer whatever. In order
that these numbers shall be multiples of n it is necessary and
sufficient that adp shall be a multiple of dv; that is, that alu
shall be a multiple of v; that is, that a shall be a multiple of
v, since y and v are relatively prime. Writing a= 6v we have
as the multiples in question the set adljv where 5 is an arbitrary
integer. This proves the theorem for the case of two numbers;
for dyv is evidently the least common multiple of in and n.
We shall now extend the proposition to any number of
integers n, n, p, q,.... The multiples in question must
be common multiples of m and n and hence of their least common
multiple A. Then the multiples must be multiples of, and p
and hence of their least common multiple /mi. But Al is evidently the least common multiple of m, n, p. Continuing in a
similar manner we may show that every multiple in question
is a multiple of aj, the least common multiple of m, n, p, q,.
And evidently every such number is a multiple of each of the
numbers i, n, p, q,..
Thus the proof of the theorem is complete.
When the two integers in and n are relatively prime their
greatest common divisor is i and their least common multiple
is their product. Again if p is prime to both mi and n it is prime
to their product mrn; and hence the least common multiple
of in, n, p is in this case mnp. Continuing in a similar manner
we have the theorem:
II. The least common multiple of several integers, priize
each to each, is equal to their product.
EXERCISES
I. In order that a common multiple of n numbers shall be the least, it is necessary and sufficient that the quotients obtained by dividing it successively by the
numbers shall be relatively prime.
2. The product of n numbers is equal to the product of their least common
multiple by the greatest common divisor of their products in — at a time.
3. The least common multiple of n numbers is equal to any common multiple M divided by the greatest common divisor of the quotients obtained on
dividing this common multiple by each of the numbers.
4. The product of n numbers is equal to the product of their greatest common
divisor by the least common multiple of the products of the numbers taken n-i
at a time.