How would I express this as a linear combination? I want to express 1 as a linear combination of 51781 and 4655.
I have a lot of other problems that consist of finding a linear combination but I just need to know do one so then I will be able to do them all. 
What steps do I need to take to understand how to solve this?

Would this be my answer?
51781 = 11*4655+576
4655 = 8*576+47
576 = 12*47+12
47 = 3*12+11
12 = 1*11+1
11= 1*11+0
 A: Hint: Use 
If $a = bq + r$, 
$a,b,q,r \in \mathbb Z, a \not = 0$ and $b \not = 0$ then 
$\gcd(a,b) = \gcd(b,r)$
Which will expand to:
\begin{align*}
    a &= bq_1 + r_1 && 0 \leq r_1 < b\\
    b &= r_1q_2 + r_2 && 0 \leq r_2 < r_1\\
    r_1 &= r_2q_3 + r_3 && 0 \leq r_3 < r_2\\
    r_2 &= r_3q_4 + r_4 && 0 \leq r_4 < r_3\\
    & \vdots && \vdots
  \end{align*}
So for your problem:
\begin{align*}
51781 &= 4655(11) + 576\\
4655 &= 576(8) + 47\\
\dots
\end{align*}
Once you get to the end you can just substitute back in the steps:
\begin{align*}
1 &= 12 - 11\\
&= (576 - 12 \cdot 47) - (47 - 3 \cdot 12)\\
&= 576 - 12 \cdot 47 + 3 \cdot 12\\
&= 576 - 13 \cdot 47 + 3 ( 576 - 12 \cdot 47)\\
&= 4(576) - 49(47)\\
&= 4(51781 - 11\cdot4655) - 49(4655 - 8\cdot576)\\
&= 4(51781) - 93(4655) + 392(576)\\
&= 4(51781) - 93(4655) + 392(51781 - 11 \cdot 4655)\\
1 &= 51781(396) + 4655(-4405)
\end{align*}
So that is your linear combination. All you have to do is substitute the steps you did to find the gcd and you will have the linear combination.
Which is the euclidean algorithm: http://en.wikipedia.org/wiki/Euclidean_algorithm
A: You should use the Extended Euclidean algorithm
A: Hint: find gcd of both, and use Euclidean algorithm
A: As the other users mentioned, the key is the Euclidean algorithm. But I will add the following: if you have two integers $x,y$, and you're wondering if for a fixed integer $c$ there exists a linear combination of $x$ and $y$ such that $ax+by=c$, with $a,b$ integers, then this will happen if and only $\gcd(x,y)$ divides $c$. 
This result has the following consequence: once you have found integers $a,b$ such that $51781a+4655b=1$, then you also know that 
$$c=c\cdot 1=c(51781a+4655b)=51781(ca)+4655(cb).$$
