# 1.9: Blankinship's Method

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In an article in the August-September 1963 issue of the American Mathematical Monthly, W.A. Blankinship$$^{1}$$ gave a simple method to produce the integers $$s$$ and $$t$$ in Bezout’s Lemma and at the same time produce $$\gcd(a,b)$$:

Given $$a>b>0$$ we start with the array $\begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \end{bmatrix}\nonumber$ Then we continue to add multiples of one row to another row, alternating choice of rows until we reach an array of the form $\begin{bmatrix} 0 & x_1 & x_2 \\ d & y_1 & y_2 \end{bmatrix}\nonumber$ or $\begin{bmatrix} d & y_1 & y_2 \\ 0 & x_1 & x_2 \end{bmatrix}\nonumber$ Then $$d=\gcd(a,b)=y_1a+y_2b$$. [The goal is to get a $$0$$ in the first column.]

Example $$\PageIndex{1}$$

First take $$a=35$$, $$b=15$$. $\begin{bmatrix} 35 & 1 & 0 \\ 15 & 0 & 1 \end{bmatrix}\nonumber$ Note $$35=15\cdot 2+5$$, hence $35+15(-2)=5.\nonumber$ So we multiply row $$2$$ by $$-2$$ and add it to row $$1$$, getting $\begin{bmatrix} 5 & 1 & -2 \\ 15 & 0 & 1 \end{bmatrix}\nonumber$ Now $$3\cdot 5=15$$ or $$15+(-3)5=0$$, so we multiply row $$1$$ by $$-3$$ and add it to row $$2$$, getting $\begin{bmatrix} 5 & 1 & -2 \\ 0 & -3 & 7 \end{bmatrix}.\nonumber$ Now we can say that $\boxed{\gcd(35,15)=5}\nonumber$ and $\boxed{5=1\cdot 35+(-2)\cdot 15.}\nonumber$

Let’s now consider a more complicated example: Take $$a=1876$$, $$b=365$$. $\begin{bmatrix} 1876 & 1 & 0 \\ 365 & 0 & 1 \end{bmatrix}\nonumber$ Now $$1876=365\cdot5+51$$ so we add $$-5$$ times the second row to the first row, getting: $\begin{bmatrix} 51 & 1 & -5 \\ 365 & 0 & 1 \end{bmatrix}\nonumber$ Now $$365=51\cdot7+8$$, so we add $$-7$$ times row $$1$$ to row $$2$$, getting: $\begin{bmatrix} 51 & 1 & -5 \\ 8 & -7 & 36 \end{bmatrix}\nonumber$ Now $$51=8\cdot 6+3$$, so we add $$-6$$ times row $$2$$ to row $$1$$, getting: $\begin{bmatrix} 3 & 43 & -221 \\ 8 & -7 & 36 \end{bmatrix}\nonumber$ Now $$8=3\cdot 2+2$$, so we add $$-2$$ times row $$1$$ to row $$2$$, getting: $\begin{bmatrix} 3 & 43 & -221 \\ 2 & -93 & 478 \end{bmatrix}\nonumber$ Then $$3=2\cdot 1+1$$, so we add $$-1$$ times row $$2$$ to row $$1$$, getting: $\begin{bmatrix} 1 & 136 & -699 \\ 2 & -93 & 478 \end{bmatrix}\nonumber$ Finally, $$2=1\cdot 2$$ so if we add $$-2$$ times row $$1$$ to row $$2$$ we get: $\label{eq:1}\begin{bmatrix} 1 & 136 & -699 \\ 0 & -365 & 1876 \end{bmatrix}.$ This tells us that $\boxed{\gcd(1876,365)=1}\nonumber$ and $\label{eq:2}\boxed{1=136\cdot 1876+(-699)365.}$ Note that it was not necessary to compute the last two entries $$-365$$ and $$1876$$ in $$\eqref{eq:1}$$. It is a good idea however to check that equation $$\eqref{eq:2}$$ holds. In this case we have: \begin{aligned} 136\cdot 1876 &= \:255136 \\ \underline{(-699)\cdot 365} &=\underline{-255135} \\ \quad &\quad 1\end{aligned} So it is correct.

### Why Blankinship’s Method works:

Note that just looking at what happens in the first column you see that we are just doing the Euclidean Algorithm, so when one element in column $$1$$ is $$0$$, the other is, in fact, the $$\gcd$$. Note that at the start we have $\begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \end{bmatrix}\nonumber$ and \begin{aligned} a &=1\cdot a+0\cdot b \\ b &=0\cdot a+1\cdot b.\end{aligned} One can show that at every intermediate step $\begin{bmatrix} a_1 & x_1 & x_2 \\ b_1 & y_1 & y_2 \end{bmatrix}\nonumber$ we always have \begin{aligned} a_1 &=x_1a+x_2b \\ b_1 &=y_1a+y_2b,\end{aligned} and the result follows. I will omit the details.

Exercise $$\PageIndex{1}$$

Use Blankinship’s method to compute the $$s$$ and $$t$$ in Bezout’s Lemma for each of the following values of $$a$$ and $$b$$.

1. $$a=267$$, $$b=112$$
2. $$a=216$$, $$b=135$$
3. $$a=11312$$, $$b=11321$$

Exercise $$\PageIndex{2}$$

Show that if $$1=as+bt$$ then $$\gcd(a,b)=1$$.

Exercise $$\PageIndex{3}$$

Find integers $$a$$, $$b$$, $$d$$, $$s$$, $$t$$ such that all of the following hold

1. $$a>0$$, $$b>0$$,
2. $$d=sa+tb$$, and
3. $$d\ne\gcd(a,b)$$.

Note that $$d$$ in Exercise 9.3 cannot be $$1$$ by Exercise 9.2.

## Footnotes

[1] Thanks to Chris Miller for bringing this method to my attention.