TI 30Xa Algorithms: Greatest Common Divisor

TI 30Xa Algorithms: Greatest Common Divisor

To find the greatest common divisor between two positive integers U and V:

Let U ≥ V. Let U = A * V + R where A is the quotient of U / V and R is the remainder. If R≠0, then V becomes the new U and R become the new V. The process repeats until R=0. At that point the value of V prior to the last calculation is the greatest common divisor (GCD) of U and V.

Example:

gcd(166, 78)

U = 166, V = 78

Algorithm Loop:

1. A = int(U / V)

2. R = U – V * int(U / V)

	A	R	U	V
Start	n/a	n/a	166	78
A = int(166 / 78) = 2 R = 166 – 2 * 78 = 10	2	10	78	10
A = int(78 / 10) = 7, R = 78 – 7 * 10 = 8	7	8	10	8
A = int(10 / 8) = 1 R = 10 – 1 * 8 = 2	1	2	8	2
A = int(8 / 2) = 4 R = 8 – 4 * 2 = 0	4	0 *STOP*

Procedure

1. Store the greater of the two numbers in memory register 1: [ STO ] [ 1 ].

2. Store the lesser of the two numbers in memory register 2: [ STO ] [ 2 ].

3. Divide memory register 1 by memory register 2. Store the integer part (no fractional part) in memory register 3: [ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ], (integer part) [ STO ] [ 3 ]

4. Figure the remainder and store the result in memory 3: [ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] [ STO ] [ 3 ]

5. If the remainder is 0, stop. The GCD is stored in memory 2.

6. If the remainder is non-zero, then store memory 2 into memory 1 then memory 3 into memory 2. You need to do it in this order. [ RCL ] [ 2 ] [ STO ] [ 1 ], [ RCL ] [ 3 ] [ STO ] [ 2 ]. Go back to Step 3 and repeat.

Examples

Example 1: GCD(26, 14)

M1 = 26, M2 = 14

	M1	M2	M3
26 [ STO ] [ 1 ], 14 [ STO ] [ 2 ]	26	14
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 1.857142857 1 [ STO ] [ 3 ]	26	14	1
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 12 [ STO ] [ 3 ] R is not zero, so we continue.	26	14	12
[ RCL ] [ 2 ] [ STO ] [ 1 ], [ RCL ] [ 3 ] [ STO ] [ 2 ]	14	12	12
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 1.166666667 1 [ STO ] [ 3 ]	14	12	1
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 2 [ STO ] [ 3 ] R is not zero, so we continue.	14	12	2
[ RCL ] [ 2 ] [ STO ] [ 1 ], [ RCL ] [ 3 ] [ STO ] [ 2 ]	12	2	2
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 6 6 [ STO ] [ 3 ]	12	2	6
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 0 [ STO ] [ 3 ] R is zero, so we stop. GCD: [ RCL ] [ 2 ]: GCD(26, 14) = 2	12	2	0

Example 2: GCD(27, 15)

M1 = 27, M2 = 15

	M1	M2	M3
27 [ STO ] [ 1 ], 15 [ STO ] [ 2 ]	27	15
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 1.8 1 [ STO ] [ 3 ]	27	15	1
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 12 [ STO ] [ 3 ] R is not zero, so we continue.	27	15	12
[ RCL ] [ 2 ] [ STO ] [ 1 ], [ RCL ] [ 3 ] [ STO ] [ 2 ]	15	12	12
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 1.25 1 [ STO ] [ 3 ]	15	12	1
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 3 [ STO ] [ 3 ] R is not zero, so we continue.	15	12	3
[ RCL ] [ 2 ] [ STO ] [ 1 ], [ RCL ] [ 3 ] [ STO ] [ 2 ]	12	3	3
[ RCL ] [ 1 ] [ ÷ ] [ RCL ] [ 2 ] [ = ] Result: 4 4 [ STO ] [ 3 ]	12	3	4
[ RCL ] [ 1 ] [ - ] [ RCL ] [ 2 ] [ × ] [ RCL ] [ 3 ] [ = ] Result: 0 [ STO ] [ 3 ] R is zero, so we stop. GCD: [ RCL ] [ 2 ]: GCD(27, 15) = 3	12	3	0

I hope you find this useful. What I hope to do with the monthly series is to demonstrate various calculations with the TI-30Xa.

Note: For June and July 2024, I will be posting on Saturdays only. I plan to resume the Saturday-Sunday schedule in August.

Eddie

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