## ABACUS: MYSTERY OF THE BEAD

The Bead Unbaffled - An Abacus Manual

Find the GCD and LCM of two numbers** Before starting these techniques it's important to understand basic

Japanese soroban operations**

GCD:Thegreatest common divisor(gcd) is also known as thegreatest common factor(gcf) and thehighest common factor(hcf). The GCD of two numbers is the largest factor that can divide into both numbers. (Note: if the GCD of two numbers equals 1, they are said to becoprime.)The

GCDfor 12 and 18

Prime factors for 12 are:x 2 x2Prime factors for 18 are:3

x2x 33

Prime factors shared by 12 & 18 areand23

Thegreatest common divisorfor 12 and 18 is6

LCM:Theleast common multipleis sometimes also known as thelowest common multiple(lcm). The LCM of two numbers is the smallest number (not zero) that is a multiple of both numbers.The

LCMfor 12 and 18

Multiples of 12 are: 0, 12, 24,, 48, 60,36...72

Multiples of 18 are: 0, 18,, 54,3690, 108...72,

Both 12 and 18 share the multiples36,& ...72

But thelowest common multipleshared by both is36Fortunately there is a very simple and efficient way to solve both GCD and LCM for two numbers. The following soroban techniques were shown to me by Enriello and by Edvaldo Siqueira both of whom are members of the Soroban / Abacus

newsgroup. To find the GCD and LCM oftwonumbers (a, b) the techniques work like this:

- Set the two numbers (a, b) onto the left side of soroban.
- Set the first number (a) one more time to the right.
- Working on the right, multiply a*b and add the product to the frame.
- Using
Euclid's algorithm*,find the GCD of a, b on the left.- Use the product calculated in step 3 to find the LCM. LCM = a*b / GCD.
*Euclid's algorithm is easily performed on an abacus. Start by subtracting the smaller number from the larger as many times as possible. In the event of a remainder, turn the process around and subtract the new smaller number from the larger. This process continues back and forth until only one number remains. The remaining number is the GCD. (Note: If at any point the numbers are radically different in size, rather than subtraction, division may be a better option.)

Example 1: Find the GCD and LCM for 12 & 18Step 1: Set 12 on rods AB and 18 on rods DE. Set another 12 on rods JK. (Fig.1)

Fig. 1

`Step 1`

A B C D E F G H I J K L M N O . . . . .1 2 0 1 8 0 0 0 0 1 2 0 0 0 0

Step 2: Multiply 12 on rods JK by 18 on DE. Add the product 216 to rods LMN. (Fig.2)

Fig. 2

Step 2

A B C D E F G H I J K L M N O . . . . .1 2 0 1 8 0 0 0 0 0 0 2 1 6 0

Find the GCD of 12 and 18 using Euclid's algorithmStep 3: Subtract 12 from 18 on rods DE. The result is 6. (Fig.3)

Fig. 3

Step 3A B C D E F G H I J K L M N O . . . . .1 2 0 0 6 0 0 0 0 0 0 2 1 6 0

Step 4 & theGCD: Turn the process around and subtract 6 from 12 on rods AB. This can be done twice resulting in zero. Since there can be no more subtractions, we know thegreatest common divisorfor the problem is 6 on rod E. (Fig.4)

Fig. 4

Step 4

A B C D E F G H I J K L M N O . . . . .0 0 0 0 6 0 0 0 0 0 0 2 1 6 0

Now find the LCM by dividing 216 on rodsLMNby 6 on rodE.Step 5 & the

LCM: Divide 6 on rod E into 216 on rods LMN. With this last step we know theleast common multiplefor 12 & 18 is the quotient answer 36 on rods KL. (Fig.5)

Fig. 5

Step 5A B C D E F G H I J K L M N O . . . . .0 0 0 0 6 0 0 0 0 0 3 6 0 0 0 .<- * * Kojima teaches; move the decimal 1 place + 1 to the left for every whole number in a divisor.

Answers: The GCD of 12 and 18 = 6 : The LCM of 12 and 18 = 36

Example 2: Find the GCD and LCM for 39 & 91Step 1: Set 39 on rods AB and 91 on rods DE. Set another 39 on rods JK. (Fig.6)

Fig. 6

`Step 1`

A B C D E F G H I J K L M N O . . . . .3 9 0 9 1 0 0 0 0 3 9 0 0 0 0

Step 2: Multiply 39 on rods JK by 91 on DE. Add the product 3549 to rods KLMN. (Fig.7)

Fig. 7

Step 2

A B C D E F G H I J K L M N O . . . . .3 9 0 9 1 0 0 0 0 0 3 5 4 9 0

Find the GCD of 39 and 91 using Euclid's algorithmStep 3: Subtract 39 rom 91 on rods DE. This can be done twice resulting in 13. (Fig.8)

Fig. 8

Step 3

A B C D E F G H I J K L M N O . . . . .3 9 0 1 3 0 0 0 0 0 3 5 4 9 0

Step 4 & the

GCD: Turn the process around and subtract 13 from 39 on rods AB. This can be done three times resulting in zero. Since there can be no more subtractions, we know thegreatest common divisorfor the problem is 13 on rods DE. (Fig.4)

Fig. 9

Step 4

A B C D E F G H I J K L M N O . . . . .0 0 0 1 3 0 0 0 0 0 3 5 4 9 0

Now find the LCM by dividing 3549 on rodsKLMNby 13 on rodDE.Step 5 & the

LCM: Divide 13 on rods DE into 3549 on rods KLMN. With this last step we know theleast common multiplefor 39 & 91 is the quotient answer 273 on rods IJK. (Fig.5)

Fig. 10

Step 5A B C D E F G H I J K L M N O . . . . .0 0 0 1 3 0 0 0 2 7 3 0 0 0 0 .<- - * * Kojima teaches; move the decimal 1 place + 1 to the left for every whole number in a divisor.

Answers: The GCD of 39 and 91 = 13 : The LCM of 39 and 91 = 273

Check your work with GCD and LCM Calculators## LCM @ math is fun

&

GCD @ math is fun

REFERENCES:

Wikipedia

Euclidean algorithm

Greatest Common Divisor

Least Common Multiple

SOROBAN TECHNIQUES:

Kojima, Takashi.

The Japanese Abacus: Its Use and Theory

Tokyo: Charles E. Tuttle, 1954

Tani, Yukio

The Magic Calculator: the way of the abacus

Tokyo: Japan Publications Trading Co, 1964

THANKS TO:

Edvaldo Siquiera

Enriello

Gary Flom

Euclid of Alexandria(ca. 325 BC-265 BC)

Members of the Soroban / Abacus Newsgroup

http://groups.yahoo.com/group/SorobanAbacus/▪

Abacus: Mystery of the Bead

▪Advanced Abacus Techniques

Print Page (.pdf format,161kb)

© April, 2006

Totton Heffelfinger Toronto Ontario Canada

totton[at]idirect[dot]com