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National and Regional Contests
Russia Contests
Moscow Mathematical Olympiad
1935 Moscow Mathematical Olympiad
021
021
Part of
1935 Moscow Mathematical Olympiad
Problems
(1)
MMO 021 Moscow MO 1935 lcm , gcd of 2 and 3 numbers
Source:
7/17/2019
Denote by
M
(
a
,
b
,
c
,
.
.
.
,
k
)
M(a, b, c, . . . , k)
M
(
a
,
b
,
c
,
...
,
k
)
the least common multiple and by
D
(
a
,
b
,
c
,
.
.
.
,
k
)
D(a, b, c, . . . , k)
D
(
a
,
b
,
c
,
...
,
k
)
the greatest common divisor of
a
,
b
,
c
,
.
.
.
,
k
a, b, c, . . . , k
a
,
b
,
c
,
...
,
k
. Prove that:a)
M
(
a
,
b
)
D
(
a
,
b
)
=
a
b
M(a, b)D(a, b) = ab
M
(
a
,
b
)
D
(
a
,
b
)
=
ab
,b)
M
(
a
,
b
,
c
)
D
(
a
,
b
)
D
(
b
,
c
)
D
(
a
,
c
)
D
(
a
,
b
,
c
)
=
a
b
c
\frac{M(a, b, c)D(a, b)D(b, c)D(a, c)}{D(a, b, c)}= abc
D
(
a
,
b
,
c
)
M
(
a
,
b
,
c
)
D
(
a
,
b
)
D
(
b
,
c
)
D
(
a
,
c
)
=
ab
c
.
least common multiple
greatest common divisor
number theory