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4.3. MULTIPLICATIVE INVERSES 59
(2) Find gcd(79, 201).
q
r 201
2
79
1
43
1
36
5
7
7
1 0
gcd(79, 201) = 1.
(3) Find gcd(182, 217).
q
r 182
1
35
5
7 0
gcd(217, 182) = 5.
⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄ ⋄
This process is called the Euclidean Algorithm, 3 and the general rule is that
at any stage of four numbers in a triangle
a
q
b
we have q = a ÷ b, the quotient of the division, and r = a%b, the remainder.
Even for large numbers the Euclidean Algorithm takes relatively few steps. 4
r
,
Example: Compute gcd(191, 156)
q
r 191
1
156
4
35
2
16
5
3
3
1 0
gcd(191, 156) = 1.
⋄
Being that this is math book, we should justify the algorithm, that is, explain
how we know it always works. Fortunately this is easy. We know that gcd(a, b) =
gcd(b, a%b), so by letting r = a%b, each consecutive pair of numbers in the “r
line” has the same gcd. Since a > b > r ≥ 0, the values in the r line are
positive but shrinking, so the algorithm must end. And when it does it, with
a 0, because gcd(r, 0) = r the final non-zero entry equals the gcd of any two
consecutive values in the line, in particular, the first two values in the line.
4.3 Multiplicative Inverses
For us to fully understand the multiplicative ciphers, decimation and linear, we
need to know how to get from enciphering key to deciphering key. An extension
3 An algorithm is a step-by-step procedure that solves a particular problem or produces
some desired outcome. The word comes from the name of Mohammed ibn-Muse al-Khwarizmi,
a mathematician in the royal court of Bagdad c. 800 A.D. Algebra likely also comes from his
name.
4 A theorem named for Gabriel Lamé, a French engineer, physicist and mathematician, says
that the number of divisions needed to find the greatest common divisor of two numbers is
no more than five times the number of decimal digits in the smaller of the two numbers.