Integer Division with Scientific Calculators

Integer Division with Scientific Calculators

Integer Division

A popular function of European scientific and fraction-featured calculators is the integer division function (also known as Euclidean division).  Integer division between two numbers (typically positive integers) return both the quotient and the remainder.

Example:  

456 int÷ 78 returns a quotient of 5 and a remainder of 66.

Here's a way to execute integer division with various calculators.

TI-36X Pro and TI-30X Pro Math Print

Integer division can be easily accomplished by using two functions:

Quotient:  Either int(x/y) or iPart(x/y)

Remainder:  mod(x,y)

Example:   456 int÷ 78

int(456/78) = 5,  mod(456, 78) = 66

Casio fx-991EX and Casio fx-991CW

Assuming x and y are positive integers, what we do depends on whether y is prime or composite.  We will assume that MathIO mode is activated.  

If y is prime:  We can divide x ÷ y, and change the result into a mixed format.

Example:  6140 int÷ 47

47 is prime.

Key in 6140 ÷ 47.  Change the result into a mixed format.  The result is 130 30/47.

Then the quotient is 130, the remainder is 47.

If y is not prime, we will not be able to do this because the fractions are always simplified to the irreducible form.

Then we will have to use the following steps:

1.   Divide x by y.   Take note of the integer portion (you may have to change the format to mixed fraction or decimal format first).   

2.  For the remainder, execute this formula:   x - quotient * y.  

Example:  5518 int÷ 32

32 is not prime.

1.  Calculate the quotient:  5518 ÷ 32 = 172 1/4.  Note that the integer part, 172 is the quotient.

2.  Calculate the remainder:  5518 - 172 * 32 = 14.  

Then the quotient is 172, the remainder is 14.  

Hewlett Packard HP 41C/Swiss Micros DM41X

Program Code:

01  LBL ᵀ INTD

02  x<>y

03  STO Z

04  x<>y

05  ST/ Z

06  MOD

07  x<>y

08  INT

09  x<>y

10  END

Eddie

All original content copyright, © 2011-2023.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Fun with the HP 20S (April 17, 2021 Edition)

Fun with the HP 20S (April 17, 2021 Edition)

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Integer Division

The program calculates the quotient and remainder of the division: 

a ÷ b

Instructions:

Enter a, press [ XEQ ] ( F ), enter b.   Get the quotient, press [ R/S ], get the remainder

Program:

01 61,41,F         LBL F

02 21,2 STO 2

03 45        ÷

04 26        R/S

05 21, 1    STO 1

06 74          =

07 21,0      STO 1

08 51,45    IP

09 26          R/S

10 22,0      STO 0

11 61,45    FP

12 55        ×

13 22,1      RCL 1

14 74        =

15 51,33,0         FIX 0

16 61,44    RND   (round)

17 61,34    ALL

18 61,26    RTN

FIX 0, RND, ALL:  round the number in the display to the nearest integer

Examples:

74 ÷ 6  = 12 R 2

Keystrokes:  74, XEQ F, 6, R/S (12) R/S (2)

1576 ÷ 237 = 6 R154

Day Number

This calculates the day number given the month and the date.   A 365-day year is assumed.  

Instructions:

1.  Store month in memory register 1

2.  Store day in memory register 2

3.  Press XEQ A

Program:

01 61,41,A         LBL A

02 22,1      RCL 1

03 31        INPUT

04 2          2

05 61,42    x≤y?

06 51,41,1         GTO 1

07 33          (

08 33        (

09 22,1      RCL 1

10 75        +

11 1          1

12 34          )

13 55        ×

14 3            3

15 0          0

16 73        .

17 6          6

18 34          )

19 51,45    IP

20 65        -

21 6            6

22 3          3

23 74          =

24 21,3      STO 3

25 51,41,2         GTO 2

26 61,41,1         LBL 1

27 33          (

28 33        (

29 22,1      RCL 1

30 65        -

31 1          1

32 34      )

33 55        ×

34 6            6

35 3          3

36 45        ÷

37 2          2

38 34        )

39 51,45    IP

40 74        =

41 21,3      STO 3

42 61,41,2         LBL 2

43 75        +

44 22,2      RCL 2

45 74        =

46 21,3      STO 3

47 61,26    RTN

Examples:

February 16:  2 STO 1, 16 STO 2, XEQ A.  Result:  47

August 22:  8 STO 1, 22 STO 2, XEQ A.   Result:  234

Source:

Duffett-Smith, Peter.  Practical Astronomy With Your Calculator  2nd Ed.  Cambridge University Press: Cambridge.  1979  ISBN 0-521-28411-2

Astronomy:  Geocentric Parallax

The program calculates the quantities ρ sin θ' and ρ cos θ' where:

ρ = distance the observer is from the center of the Earth in meters

θ' = geocentric latitude in degrees

Inputs:

h = height of observer above sea level in meters

L = latitude of the observer in degrees, north is positive

Calculations:

h' = h / 6378140   (estimated radius of Earth is about 6,378 km)

u = atan(0.996647 tan L)

ρ sin θ' = 0.996647 sin u + h' sin L

ρ cos θ' = cos u + h' cos L

Program:

01 61,41,b         LBL B

02 61,23    DEG

03 33          (

04 22,1      RCL 1

05 25        TAN

06 55        ×

07 73        .

08 9          9

09 9          9

10 6            6

11 6          6

12 4          4

13 7          7

14 21,0      STO 0

15 34          )

16 51,25    ATAN

17 21,3      STO 3

18 22,2      RCL 2

19 45        ÷

20 6            6

21 3            3

22 7            7

23 8          8

24 1            1

25 4          4

26 0          0

27 74          =

28 21,4        STO 4

29 22,0      RCL 0

30 55          ×

31 22,3      RCL 1

32 23        SIN

33 75        +

34 21,1      RCL 1

35 23          SIN

36 55            ×

37 22, 4        RCL 4

38 74        =

39 26          R/S

40 22, 3        RCL 3

41 24            COS

42 75          +

43 22, 4      RCL 4

44 55        ×

45 22, 1      RCL 1

46 24          COS

47 74          =

48 61,26      RTN

Registers used:

R1 = L  (input)

R2 = h   (input)

R3 = u

R4 = h / 6378140

R0 = 0.996647

Example:

Input:

L = 45.76° N

h = 53.48 m

54.76 STO 1, 53.48 STO 2, XEQ B

Result:  

ρ sin θ' = .712859241, R/S

ρ cos θ' = .698873028

Source:

Duffett-Smith, Peter.  Practical Astronomy With Your Calculator  2nd Ed.  Cambridge University Press: Cambridge.  1979  ISBN 0-521-28411-2

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.