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Putnam
1974 Putnam
A6
A6
Part of
1974 Putnam
Problems
(1)
Putnam 1974 A6
Source: Putnam 1974
5/28/2022
Given
n
n
n
, let
k
=
k
(
n
)
k = k(n)
k
=
k
(
n
)
be the minimal degree of any monic integral polynomial
f
(
x
)
=
x
k
+
a
k
−
1
x
k
−
1
+
…
+
a
0
f(x)=x^k + a_{k-1}x^{k-1}+\ldots+a_0
f
(
x
)
=
x
k
+
a
k
−
1
x
k
−
1
+
…
+
a
0
such that the value of
f
(
x
)
f(x)
f
(
x
)
is exactly divisible by
n
n
n
for every integer
x
.
x.
x
.
Find the relationship between
n
n
n
and
k
(
n
)
k(n)
k
(
n
)
. In particular, find the value of
k
(
n
)
k(n)
k
(
n
)
corresponding to
n
=
1
0
6
.
n = 10^6.
n
=
1
0
6
.
Putnam
algebra
polynomial