In number theory, two positive integers *a* and *b* are said to be **multiplicatively independent** if their only common integer power is 1. That is, for all integer *n* and *m*,
a
n
=
b
m
implies
n
=
m
=
0
. Two integers which are not multiplicatively independent are said to be multiplicatively dependent.

For example, 36 and 216 are multiplicatively dependent since
36
3
=
(
6
2
)
3
=
(
6
3
)
2
=
216
2
and 6 and 12 are multiplicatively independent

Being multiplicatively independent admits some other characterizations. *a* and *b* are multiplicatively independent if and only if
log
(
a
)
/
log
(
b
)
is irrational. This property holds independently of the base of the logarithm.

Let
a
=
p
1
α
1
p
2
α
2
⋯
p
k
α
k
and
b
=
q
1
β
1
q
2
β
2
⋯
q
l
β
l
be the canonical representations of *a* and *b*. The integers *a* and *b* are multiplicatively dependent if and only if *k* = *l*,
p
i
=
q
i
and
α
i
β
i
=
α
j
β
j
for all *i* and *j*.

Büchi arithmetic in base *a* and *b* define the same sets if and only if *a* and *b* are multiplicatively dependent.

Let *a* and *b* be multiplicatively dependent integers, that is, there exists *n,m>1* such that
a
n
=
b
m
. The integers *c* such that the length of its expansion in base *a* is at most *m* are exactly the integers such that the length of their expansion in base *b* is at most *n*. It implies that computing the base *b* expansion of a number, given its base *a* expansion, can be done by transforming consecutive sequences of *m* base *a* digits into consecutive sequence of *n* base *b* digits.