How Do You Find the Multiplicative Inverse in \( Z_{360}[x] \)?

[corkycorey101]

How would you find the multiplicative inverse of the following?

Let $R=Z_{360}[x]$
Find the multiplicative inverse of $f(x)=30x^4+120x^2+240x+7$ in $R$.

Do you have to solve it using the Euclidean Algorithm? If so, I'm not sure how to do that.
This problem has me stumped.

Any help is much appreciated.
Thanks!

 

[I like Serena]

How would you find the multiplicative inverse of the following?

Let $R=Z_{360}[x]$
Find the multiplicative inverse of $f(x)=30x^4+120x^2+240x+7$ in $R$.

Do you have to solve it using the Euclidean Algorithm? If so, I'm not sure how to do that.
This problem has me stumped.

Any help is much appreciated.
Thanks!

Hi corkycorey101! Welcome to MHB! ;)

Let the multiplicative inverse be $f^{-1}(x)=a_n x^n + ... + a_0$.
Then:
$$7a_0 \equiv 1 \pmod{360}\\
240a_0 + 7a_1 \equiv 0 \pmod{360}\\
...
$$

To solve $7a_0 \equiv 1$, we could apply the Euclidean algorithm, but let's try inspection first.
If we divide $360 + 1$ by $7$, the remainder is $4$.
If we divide $2\cdot 360 + 1$ by $7$, the remainder is $0$.
So:
$$7^{-1} \equiv \frac{2\cdot 360 + 1}{7} \equiv 103 \pmod{360}$$
Generally, small as $7$ is, we can be sure to find sufficient information within 3 tries.