# Existence proof

• Sep 22nd 2010, 12:46 PM
paulrb
Existence proof
Show that for every integer a not divisible by 11 there is another integer b with
a*b == 1 (mod 11).

This problem has me stumped...
• Sep 22nd 2010, 12:55 PM
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Quote:

Originally Posted by paulrb
Show that for every integer a not divisible by 11 there is another integer b with
a*b == 1 (mod 11).

This problem has me stumped...

Actually there are infinite. The problem stated another way: show that if $\displaystyle a \not \equiv 0\pmod{11}$ then a has a multiplicative inverse mod 11. It follows immediately from fermat's little theorem / euler's theorem.
• Sep 22nd 2010, 01:29 PM
paulrb
Thanks...I should have specified though, I've only been in this number theory class for 3 weeks and we have only covered basic theorems about congruence / number theory. Nothing with fermat's little theorem / euler's theorem so I don't think it would be acceptable for me to use that in a proof.
• Sep 22nd 2010, 02:05 PM
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Quote:

Originally Posted by paulrb
Thanks...I should have specified though, I've only been in this number theory class for 3 weeks and we have only covered basic theorems about congruence / number theory. Nothing with fermat's little theorem / euler's theorem so I don't think it would be acceptable for me to use that in a proof.

I think we can do as follows. It will be a stronger result than what you need to prove.

Let p be any prime, and let a be an integer not divisible by p.

Consider the set $\displaystyle S=\{0,a,2a,\,\ldots\,,(p-1)a\}$. Claim: each element of S is distinct modulo p.

Proof: Suppose the opposite. Then there exist integers x, y such that $\displaystyle x\not\equiv y\pmod{p}$ and $\displaystyle ax\equiv ay\pmod{p}$. The latter implies $\displaystyle a(x-y)\equiv 0\pmod{p}$. So we have a product of two non-multiples of p being divisible by p; contradiction.

Now consider that there are p distinct equivalence classes in S modulo p, thus it is the set of all equivalence classes modulo p, and $\displaystyle \overline{1}\in S$.
• Sep 22nd 2010, 02:44 PM
paulrb
That's a great proof, thank you!
• Sep 23rd 2010, 09:12 AM
roninpro
I'd just like to point out that a more standard approach would be to use Bezout's identity.

If 11 does not divide $\displaystyle a$, then $\displaystyle \gcd(a,11)=1$. By Bezout's identity, there exist integers $\displaystyle x,y$ such that $\displaystyle ax+11y=1$. Taking everything modulo 11, $\displaystyle ax+11y\equiv 1\pmod{11}$, and $\displaystyle ax\equiv 1\pmod{11}$. The term $\displaystyle x$ is what we are looking for.
• Sep 23rd 2010, 09:18 AM
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Quote:

Originally Posted by roninpro
I'd just like to point out that a more standard approach would be to use Bezout's identity.

If 11 does not divide $\displaystyle a$, then $\displaystyle \gcd(a,11)=1$. By Bezout's identity, there exist integers $\displaystyle x,y$ such that $\displaystyle ax+11y=1$. Taking everything modulo 11, $\displaystyle ax+11y\equiv 1\pmod{11}$, and $\displaystyle ax\equiv 1\pmod{11}$. The term $\displaystyle x$ is what we are looking for.

Thanks, I wasn't sure if Bezout's identity was available after 3 weeks of intro number theory, I should study standard progression of theorems from axioms.