# Fundamentals of Programming: Modulo arithmetic

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):

1r 3 7)1073

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
MOD = 1 DIV = 3 17 / 5
MOD = 2 DIV = 3 8 / 2
MOD = 0 DIV = 4 6 / 9
MOD = 6 DIV = 0 (9 does not divide into 6 at all!) Now try these explicit calculations:
11 MOD 8
= 3 8 MOD 4
= 0 6 DIV 5
= 1 600 DIV 20
= 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.

Note that Visual Basic does not have a DIV function and uses `A\B` instead |

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):
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
``` |