## Thursday, November 22, 2018

### TI-84 Plus and HP Prime: Chinese Remainder Theorem

TI-84 Plus and HP Prime:  Chinese Remainder Theorem

Introduction

The Chinese Remainder Theorem deals with solving the following congruences:

x ≡ r₀ mod m₀
x ≡ r₁ mod m₁
x ≡ r₂ mod m₂
...

where m₀, m₁, m₂, etc are all relatively prime.  Two integers are relatively prime when both integers have a GCD (greatest common divisor) is 1.

We are going to focus on the two congruent system:

(I)
x ≡ r mod s
x ≡ t mod u

where the solution is x mod s*u.

HP Prime Function CAS.inchinrem

To solve the Chinese Remainder Theorem, use the function inchinrem.

Syntax (reference (I) above):

Home/Programming Mode Syntax:  CAS.inchinrem([r, s], [t, u]).
CAS Mode Syntax:  inchinrem([r, s], [t, u])

The answer returned is x mod s*u in vector form [x, s*u].

Where to find inchinrem:  [Toolbox], (CAS), 5.  Integer, 7. Division, 3.  Chinese Remainder

TI-84 Plus Program CTR2

"2018-11-18 EWS"
Disp "CHINESE REMAINDER","X=R MOD S","X=T MOD U"
Prompt R,S,T,U
If gcd(S,U)≠1
Then
Disp "NO SOLUTION"
Stop
T-R→W
U*fPart(abs(W)/U)→W
If T-R<0 font="">
U-W→W
0→Y
0→N
Repeat W=N
1+Y→Y
U*fPart(S*Y/U)→N
End
S*Y+R→X
S*U→M
Disp "SOLUTION:",X,"MOD",M

Examples

Example 1:

x ≡ 3 mod 19
x ≡ 8 mod 11

Solution:  [41, 209],  41 mod 209

Example 2:

x ≡ 4 mod 14
x ≡ 7 mod 17

Solution:  [228, 238],  228 mod 238

Source:

Silverman, Joseph H.  A Friendly Introduction to Number Theory Prentice Hall, Inc: Upper Saddle River, New Jersey  2001.  ISBN 0-13-030954-0

Happy Thanksgiving!

Eddie

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