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Bezout's Identity: Proof & Examples

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

A linear combination of two integers can be shown to be equal to the greatest common divisor of these two integers. This is the essence of the Bazout identity. In this lesson, we prove the identity and use examples to show how to express the linear combination.

Bezout's Identity

Although they might appear simple, integers have amazing properties. In this lesson, we revisit an algorithm for finding the greatest common divisor of integers and then use this algorithm to explore the Bazout identity. This exploration includes some examples and a proof.

The Greatest Common Divisor

Let's make sense of the phrase greatest common divisor (gcd). There are 3 parts: divisor, common and greatest. For example, if we have the number, 120, we could ask ''Does 1 go into 120?'' Yes, 120 divided by 1 is 120 with no remainder. Thus, 1 is a divisor of 120. How about 2? Well, 120 divide by 2 is 60 with no remainder. Thus, 2 is also a divisor of 120. So is, 3, 4, 5, and 6. How about 7? In this case, 120 divided by 7 is 17 but there is a remainder (of 1). Thus, 7 is not a divisor of 120. We could do this test by division and get all the divisors of 120:


1,2,3,4,5,6,8,10,12,15,20,24,30,40,60,120


Wow! Lots of work. How about the divisors of another number, like 168? Same process of division checks for divisors with no remainder. The divisors of 168:


1,2,3,4,6,7,8,12,14,21,24,28,42,56,84,168


For 120 and 168, we have all the divisors. What are the common divisors? These are the divisors appearing in both lists:


1,2,3,4,6,8,12,24


And the ''g'' part of gcd is the greatest of these common divisors: 24. Thus, the gcd of 120 and 168 is 24.

There is a better method for finding the gcd. Take the larger of the two numbers, 168, and divide by the smaller number, 120. We get 1 with a remainder of 48. Thus, 168 = 1(120) + 48.

Divide the number in parentheses, 120, by the remainder, 48, giving 2 with a remainder of 24. Thus, 120 = 2(48) + 24.

Again, divide the number in parentheses, 48, by the remainder 24. We get 2 with a remainder of 0. Thus, 48 = 2(24) + 0.

To summarize:


168=1(120)+48;120=2(48)+24;48=2(24)+0


When the remainder is 0, we stop. The remainder, 24, in the previous step is the gcd. This method is called the Euclidean algorithm.

Bazout's Identity

The Bazout identity says for some x and y which are integers,


ax+by=gcd_of_a_and_b


For a = 120 and b = 168, the gcd is 24. Thus, 120x + 168y = 24 for some x and y. Let's find the x and y.

Start with the next to last line of the Euclidean algorithm, 120 = 2(48) + 24 and write


24=120-2(48)


In the line above this one, 168 = 1(120)+48. Thus,


48=168-1(120)


Substitute 168 - 1(120) for 48 in 24 = 120 - 2(48), and simplify:


24=120-2(48)=120-2(168-1(120))=120+2(120)-2(168)=3(120)-2(168)


Compare this to 120x + 168y = 24 and we see x = 3 and y = -2.

Checking these values:


120x+168y=120(3)+168(-2)=360-326=24


And it works!

Proof of Bazout's Identity

To prove Bazout's identity, write the equations in a more general way.


a=alpha1_b+r1


α1 is an integer and r1 is a remainder.

In the second line,


b=alpha2_r1+r2


and in the third line we see how the remainders move from line to line:


r1=alpha3_r2+r3


From a = α1 b + r1, the remainder:


r1=a-alpha1_b


r1 is a linear combination of a and b (an integer times a plus an integer times b).

On the next line, b = α2 r1 + r1:


r2=b-alpha2_r1=b-alpha2_(a-alpha1_b)=-alpha2_a+(a+alpha1_alpha2)b


And again, the remainder is a linear combination of a and b.

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