B3

Part of 1986 Putnam

Problems(1)

Source:

\Gamma

consist of all polynomials in

x

with integer coefficients. For

f

g

\Gamma

m

a positive integer, let

f \equiv g \pmod{m}

mean that every coefficient of

f-g

is an integral multiple of

m

n

p

be positive integers with

p

prime. Given that

f,g,h,r

s

\Gamma

rf+sg\equiv 1 \pmod{p}

fg \equiv h \pmod{p}

, prove that there exist

F

G

\Gamma

F \equiv f \pmod{p}

G \equiv g \pmod{p}

FG \equiv h \pmod{p^n}