Fundamentals of Programming: Modulo arithmetic

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UNIT 1 - ⇑ Fundamentals of Programming ⇑

← Fundamentals of Structured Programming Modulo arithmetic Logical bitwise operators →


Modular arithmetic is all about finding the remainder from long division (MOD), and the total number of times that a number goes into a division (DIV). Let's take a look at a quick example of 10 divided by 7 (you might want to remind yourself about long division):

    1 r 3
 7)10
    7
    3

Hopefully that wasn't too hard. We now need to introduce some terminology, MOD and DIV:

  • MOD = finds the remainder from long division i.e. 10 MOD 7 = 3
  • DIV = finds the number of divides in long division i.e. 10 DIV 7 = 1
Exercise: MOD and DIV
Try these examples, working out the MOD and DIV of each: 7 / 2

Answer :

MOD = 1 DIV = 3

17 / 5

Answer :

MOD = 2 DIV = 3

8 / 2

Answer :

MOD = 0 DIV = 4

6 / 9

Answer :

MOD = 6 DIV = 0 (9 does not divide into 6 at all!)

Now try these explicit calculations: 11 MOD 8

Answer :

= 3

8 MOD 4

Answer :

= 0

6 DIV 5

Answer :

= 1

600 DIV 20

Answer :

= 30

Hopefully you are now pretty good with MOD and DIV, but what exactly is the point of all this? A very common example in past exam paper has been using the MOD and DIV to work out a binary equivalent of a denary number.

Example: Converting Denary to Binary using DIV
sub convertDtoB(byVal base10 as integer)
  dim base2(7) as integer 'create an array to store the binary
  dim temp as integer = base10
  for i = 7 to 0 step -1 'loop through each binary bit starting from the biggest value
    base2(i) =  temp \ (2^i)  'temp DIV 2^i
    temp = temp MOD 2^i
  next
  console.write(base10 & " in binary = ")
  for i = 7 to 0 step -1 'loop through each binary bit starting from the biggest value
    console.write(base2(i))
  next
end sub

Try the code out and see if it works. Try to write a trace table and see how it works for the number 67 (again another popular question in exams):

Answer :

base10 temp 2^i i base2
0 1 2 3 4 5 6 7
67 67
128 7 0
3 64 6 1
32 5 0
16 4 0
8 3 0
4 2 0
1 2 1 1
0 1 0 1

Output:

67 in binary = 01000011

Another common use is in finding out whether a number is odd or even using the MOD function. We know that MOD returns the remainder from a division sum. So for example 4 MOD 2 = 0, 5 MOD 2 = 1, 6 MOD 2 = 0 and so on. By modding something with 2 we can work out whether it is odd or not due to the return value.

Example: Finding out if a number is Odd using MOD
Dim testNum as Integer
Dim temp as Integer
console.writeline("please insert a number to test:")
testNum = console.readline()
If testNum MOD 2 = 0 Then
  Console.Writeline("The number is even")
Else
  Console.Writeline("The number is odd")
End If