Euclid, Antanairesis, and
Greatest Common Divisors
Dr. Maurice Burke
Montana State University
MEA-MFT 2015 Billings
Conclusion
Greatest Common Divisor (gcd)
• What is the greatest common divisor of 120
and 168?
24
• What method(s) did you try for finding this
gcd?
Factor Method?
120 = 23 × 3 × 5 168 = 23 × 3 × 7
Least Common Multiple (lcm)
• Factorization Method
lcm(120,168) = 840 = 23 × 3 × 5 × 7
• Hey, is there a formula relating gcd and lcm????
840 = 23 × 3 × 5 × 7
840 =
23 ×3×5×7 23 ×3
23 ×3
840 = LCM(120,168) =
=
23 ×3×5 7×23 ×3
23 ×3
120×168
gcd(120,168)
Formula : gcd 𝑎, 𝑏 × 𝑙𝑐𝑚 𝑎, 𝑏 = 𝑎 × 𝑏
What is gcd
• Do it on the calculator?
TI-84
TI-Nspire
14 7
,
15 12
?
What is the lcm
14 7
,
15 12
?
• Use the formula?!? Does it work here?!?
14
7
×
15
12
= 𝑔𝑐𝑑 × 𝑙𝑐𝑚 =
𝑥
𝑦
So, =
• Use the Nspire:
14
3
7
60
×
𝑥
𝑦
Definitions Matter
• What is your definition of “divisor” or what do you
mean by “A is a divisor of B?”
14
15
=
𝟕
𝟔𝟎
×8
𝟕
𝟏𝟐
=
𝟕
×5
𝟔𝟎
“A is a divisor of B provided A guzinta B a whole number
of times.”
• What is your definition of “multiple” or what do you
mean by “B is a multiple of A?”
𝟏𝟒
𝟑
=
14
×
15
5
𝟏𝟒
𝟑
=
7
×
12
8
Domain matters in definitions.
Good Explorations
• Find a shortcut method for determining the
gcd
?????
?????
?????
𝑎 𝑐
,
𝑏 𝑑
and lcm
𝑎
𝑏
𝑎 𝑐
,
𝑏 𝑑
𝑐
𝑑
× = 𝑔𝑐𝑑
gcd
𝑎 𝑐
,
𝑏 𝑑
lcm
𝑎 𝑐
,
𝑏 𝑑
𝑎 𝑐
,
𝑏 𝑑
=
=
.
𝑎 𝑐
𝑙𝑐𝑚 ,
𝑏 𝑑
gcd(𝑎,𝑐)
𝑙𝑐𝑚(𝑏,𝑑)
𝑙𝑐𝑚(𝑎,𝑐)
gcd(𝑏,𝑑)
Many Entry Points For Investigating
General GCDs and LCMs
So What Can You
Say about W?
8 =W
5 =W
W is a common
multiple of the
circle and the
triangle. Is W
the LCM( , )?
What about
the GCD?
=5
=8
8
8
So,
is
8
GCD. What
about
5
?
What is gcd
• TI-NSPIRE
2, 1 ?
Euclid’s Method
“Alternating Subtraction”
168
120
Subtract 120 from heavier side.
48
120
Subtract 48 from heavier side.
48
72
Subtract 48 from heavier side.
48
24
Subtract 24 from heavier side.
24
24
Subtract 24 again.
0
24
Process ends. GCD is 24.
Why Does It Work?
Very Simple: Distributive Law
If A>B and C is a common divisor of A and B, i.e.
𝑡ℎ𝑒𝑟𝑒 𝑎𝑟𝑒 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠 𝑛 𝑎𝑛𝑑 𝑚 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡
𝐴 = 𝑛 × 𝐶 𝑎𝑛𝑑 𝐵 = 𝑚 × 𝐶
𝑇ℎ𝑒𝑛 𝐴 − 𝐵 = 𝑛 − 𝑚 × 𝐶
Therefore, C is a common divisor of A-B and B.
Likewise, if C is a common divisor of A-B and B then
C is a common divisor of A and B.
So, gcd(A,B) = gcd(A-B,B)
Euclid’s Method
“Alternating Subtraction”
168
120
Subtract 120 from heavier side.
48
120
Subtract 48 from heavier side.
48
72
Subtract 48 from heavier side.
48
24
Subtract 24 from heavier side.
24
24
Subtract 24 again.
0
24
Process ends. GCD is 24.
Shortcut: Euclidean Algorithm
If A > B and B is repeatedly subtracted from A
until the result is less than B, then that result is
the remainder you get when you use long
division to divide A by B. Furthermore, the
number of times you subtract B from A is called
the quotient.
So using division instead of repeated
subtraction, here is the shortened process:
Euclidean Algorithm:
Replace heavier side by the remainder when heavier
side is divided by the lighter side.
168
120
Quotient:
1
Remainder:
48
48
120
Quotient:
2
Remainder:
24
48
24
Quotient:
2
Remainder:
0
0
24
Process Ends
GCD = 24
Use Spreadsheets to Explore (try
finding gcd(2016, 366)
To Get Quotients
Algorithm Applied to Fractions
Algorithm applied to gcd( 2, 1)
Euclid’s Conclusion
Common Core
6.NS.
Compute fluently with multi-digit numbers and find common factors
and multiples.
4. Find the greatest common factor of two whole numbers less than or
equal to 100 and the least common multiple of two whole numbers
less than or equal to 12. Use the distributive property to express a
sum of two whole numbers 1–100 with a common factor as a multiple
of a sum of two whole numbers with no common factor. For example,
express 36 + 8 as 4 (9 + 2).
8.NS.
Know that there are numbers that are not rational, and approximate
them by rational numbers.
2. Use rational approximations of irrational numbers to compare the size
of irrational numbers, locate them approximately on a number line
diagram, and estimate the value of expressions (e.g., π/2). For example,
by truncating the decimal expansion of √2, show that √2 is between 1 and
2, then between 1.4 and 1.5, and explain how to continue on to get better
approximations.
The Curriculum
We restrict our students to the positive integers when
speaking about gcd and lcm for good reasons:
1. The gcd and lcm of an irrational number and a
rational number do not exist.
2. Although they exist in general for any set of positive
rational numbers, the gcd and lcm of frctions are
computationally more difficult and theoretically not
needed for our students.
3. The gcd and lcm are useful and easily computable,
with the prime factorization method, for the number
theory of K-12 mathematics education.
4. With technology, this situation changes!
Antanairesis: Its Historical Significance
While the process originates in the Pythagorean
study of natural numbers, it might have been
applied to magnitudes such as lengths to first
discover incommensurable lengths:
Show: Side CD and Diagonal AC of
Pentagram are Incommensurable
Ananairesis reduces the
comparison of lengths AC
and CD to comparing
lengths CD and GD. But
CD and GD are also the
sides of a golden triangle.
Thus, the antanairesis
process will continue
indefinitely. This means
AC and CD are
incommensurable.
A Greek Theory of Proportion
For any 𝑚 𝑎𝑛𝑑 𝑛 𝑛𝑎𝑡𝑢𝑟𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟𝑠,
𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒 𝑚
2=
≠ ,
𝑆𝑖𝑑𝑒 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒
𝑛
Same as saying:
𝑛 × 𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒 ≠ 𝑚 × 𝑆𝑖𝑑𝑒 𝑜𝑓 𝑆𝑞𝑢𝑎𝑟𝑒
So what would it mean, for circles, to say
𝐶1
𝐶2
=
𝑑1
?
𝑑2
Pythagorean Theory
• By their theory of proportion, Pythagoreans meant
that there existed natural numbers m and n such
𝐶1
𝑚
𝑑1
𝑚
that = 𝑎𝑛𝑑
= .
𝐶2
𝑛
𝑑2
𝑛
• This formulation of proportion requires the
diameters to be commensurable and requires the
circumferences to be commensurable. NOT GOOD.
E.G., if
𝑑1
𝑑2
=
that 𝑑1 = 𝑚
𝑚
𝑛
then 𝑛 × 𝑑1 = 𝑚 × 𝑑2 . This means
𝑑2
×
𝑛
and 𝑑2 = 𝑛
𝑑2 have a common divisor
𝑑2
.
𝑛
𝑑2
× .
𝑛
(→←)
Hence, 𝑑1 and
Aristotle Asserts
• Antanairesis can be used to define what it means
for two ratios of magnitudes to be in proportion.
• Some historians think the Greeks might have
known the following: By paying attention to the
number of times each number gets subtracted
before the balance tips in the antanairesis
process, you generate a sequence that is
characteristic to the ratio of the two numbers
being compared. If their antanairesis sequences
are the same, two ratios are “proportional.”
Antanairesis Sequence for 168 and 120
and for 6552 and 4680
Antanairesis sequence for 3, 1
Applied to π: 3+1/(7+1/(15+1/(1)))
(OK, so we are cheating.)
Ratios of magnitudes are kind of like
numbers
Omar Khayyam (1048 – 1131)
“…a ratio between magnitudes is
conjoined with something numerical or in
the potentiality of number” …its size
“should be regarded as being abstracted in
the intellect from these adjunct characters
and as being attached to number: not as a
true absolute number, for it may be that
the ratio between A and B is not
numerical…”
Isaac Newton (1643-1727)
“By Number we understand, not so much a
Multitude of Unities, as the abstracted Ratio of
any Quantity, to another Quantity of the same
Kind, which we take for Unity. And this is
threefold; integer, fracted, and surd: An Integer,
is what is measured by Unity; a Fraction, that
which a submultiple Part of Unity measures; and
a Surd, to which Unity is incommensurable.”
(Universal Arithmetick, 1707)
So, What is a real number?
The antanairesis-based definition of Proportion
appears to be displaced during the life of
Aristotle by the Eudoxian notion of Proportion
that inspires David Hilbert’s 19th Century
breakthrough definition of “real number”.
Antanairesis inspires the study of continued
fractions and methods of rational
approximations to irrational numbers.
Conclusion