Discrete Structures
Sept 26 2011
Assignment 6:
Due at beginning of class Wednesday, Oct 12
Prof. Hopcroft
*******Note: this homework is subject to the style guide on the website.
Points will be
deducted for homeworks not following the guidelines.********
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1. If, for a given
a
there exists
x
such that
ax
≡
1
mod
m
does that imply that
a
and
m
are relatively
prime? Prove why or why not.
Proof.
This implies that
x
is
a
’s inverse in
mod
m
.
We have a theorem, that says if
a
and
m
are
relatively prime, then
x
exists (Euclid’s Extended Algorithm).
Hence, we have the intuition that
we should try to prove
a
and
m
are relatively prime.
But note, we can not directly use Euclid’s
Algorithm, as it assumes what we want to prove. (Another way to phrase it, is we want to prove the
reverse direction.)
There are a few proofs that work, I present the most straight forward one:
Using the given let us manipulate it a little bit:
ax
≡
1
mod
m
ax

km
= 1
for some integer
k
ax
= 1 +
km
Now, let us look at the gcd
gcd(
ax, m
) = gcd(
km
+ 1
, m
)
.
It may seem obvious from here, but if we add a little bit of argumentation, our proof will be mathe
matically sound.
Let
d
be the gcd(
km
+ 1
, m
). Because
d
divides
m
and
d
divides
km
+ 1, this implies that
d
divides 1.
Which, can only be the case if
d
= 1.
Hence, we have shown gcd(
ax, m
) = 1, which implies no number that divides
m
divides either
a
or
x
.
Hence, there is no number (other than 1) that divides both
m
and
a
.
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 '07
 SELMAN
 Number Theory, ax, Prime number, Square number

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