36
1.96
Let
m
,
m
0
be positive integers, let
d
=
(
m
,
m
0
)
, and let
b
≡
b
0
mod
d
.
Prove that any two solutions of the system
x
≡
b
mod
m
x
≡
b
0
mod
m
0
are congruent mod
`
, where
`
=
lcm
{
m
,
m
0
}
.
Solution.
Absent.
1.97
On a desert island,
f
ve men and a monkey gather coconuts all day, then
sleep. The
f
rst man awakens and decides to take his share. He divides
the coconuts into
f
ve equal shares, with one coconut left over. He gives
the extra one to the monkey, hides his share, and goes to sleep. Later,
the second man awakens and takes his
f
fth from the remaining pile; he too
f
nds one extra and gives it to the monkey. Each of the remaining three men
does likewise in turn. Find the minimum number of coconuts originally
present.
Solution.
Here are the equations arising from the story. Let
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 Fall '11
 KeithCornell
 Division, Natural number, Prime number, Power of two

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