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8
2021 Algebra/NT #8: LCM's and GCD's
2021 Algebra/NT #8: LCM's and GCD's
Source:
May 30, 2021
number theory
Problem Statement
For positive integers
a
a
a
and
b
b
b
, let
M
(
a
,
b
)
=
lcm
(
a
,
b
)
gcd
(
a
,
b
)
,
M(a,b) = \tfrac{\text{lcm}(a,b)}{\gcd(a,b)},
M
(
a
,
b
)
=
g
c
d
(
a
,
b
)
lcm
(
a
,
b
)
,
and for each positive integer
n
≥
2
,
n \ge 2,
n
≥
2
,
define
x
n
=
M
(
1
,
M
(
2
,
M
(
3
,
…
,
M
(
n
−
2
,
M
(
n
−
1
,
n
)
)
⋯
)
)
)
.
x_n = M(1, M(2, M(3, \dots , M(n - 2, M(n - 1, n))\cdots))).
x
n
=
M
(
1
,
M
(
2
,
M
(
3
,
…
,
M
(
n
−
2
,
M
(
n
−
1
,
n
))
⋯
)))
.
Compute the number of positive integers
n
n
n
such that
2
≤
n
≤
2021
2 \le n \le 2021
2
≤
n
≤
2021
and
5
x
n
2
+
5
x
n
+
1
2
=
26
x
n
x
n
+
1
.
5x_n^2 + 5x_{n+1}^2 = 26x_nx_{n+1}.
5
x
n
2
+
5
x
n
+
1
2
=
26
x
n
x
n
+
1
.
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