1. Miniconference on the Mathematics of Computation
MTH 210
Euclidean Algorithm
Dr. Anthony Bonato
Ryerson University

2. Greatest Common Divisors
m and n integers > 1
among all the divisors of m and n, the largest is called the greatest common divisor
write gcd(m,n)

3. Examples gcd(12,14) gcd(105,385) divisors of 12: 1,2,3,4,6,12
105 = 3 x 5 x 7
385 = 5 x 7 x 11
gcd(105,385) = 5 x 7 = 35

4. Key Facts gcd(n,0) = n If a = bq + r, then gcd(a,b) = (b,r)
useful idea: to find gcd(a,b) simplifies to “smaller” case: gcd(b,r)
call r the remainder

5. Euclidean Algorithm Goal: find gcd(a,b) Step 1: Express a = bq + r
Step 2: Find gcd(b,r).
Step 3: Express b = sr + t.
Step 4: Find gcd(r,t). …
Keep going until remainder is zero.

6. Why it works uses Key Fact 2: If a = bq + r, then gcd(a,b) = (b,r)
the remainders are positive and keep getting smaller in each step
eventually remainders go down to 0, then use Key Fact 1: gcd(r,0) = r.

7. Exercises

8. Linear Diophantine Equations
Miniconference on the Mathematics of Computation
MTH 210
Linear Diophantine Equations
Dr. Anthony Bonato
Ryerson University

9. Equations with integer solutions
main question: If a, b, and c are integers, when does a solution exist and how can you find a solution to
ax + by = c,
where x and y are integers?
these are linear Diophantine equations

10. and sometimes a solution does not even exist!
sometimes trial-and-error works, but this is not always very efficient.
and sometimes a solution does not even exist!
example: 2x + 4y = 5 has no solution (why?)

11. Key Fact:
ax + by = c has a solution if and only if gcd(a, b) divides c.

12. x = x1 + tb/d, y = y1 – ta/d, where t is an integer.
How to solve ax + by = c?
Calculate gcd(a,b) by using the Euclidean Algorithm.
Does gcd(a,b) divide c?
If NO, then there is no solution to the linear Diophantine equation.
If YES then
Solve ax + by = gcd(a,b) = d by working backwards through your steps in the Euclidean Algorithm. This gives a particular solution (x1,y1)
The general solution is:
x = x1 + tb/d, y = y1 – ta/d, where t is an integer.