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National and Regional Contests
PEN Problems
PEN A Problems
69
69
Part of
PEN A Problems
Problems
(1)
A 69
Source:
5/25/2007
Prove that if the odd prime
p
p
p
divides
a
b
−
1
a^{b}-1
a
b
−
1
, where
a
a
a
and
b
b
b
are positive integers, then
p
p
p
appears to the same power in the prime factorization of
b
(
a
d
−
1
)
b(a^{d}-1)
b
(
a
d
−
1
)
, where
d
=
gcd
(
b
,
p
−
1
)
d=\gcd(b,p-1)
d
=
g
cd
(
b
,
p
−
1
)
.
modular arithmetic
number theory
prime factorization
Divisibility Theory