Integer division and remainder after division

Div

The div operation is an integer division, where the result is a number without fractional part.

The div operation calculates the integer part of the result of dividing integers (partial quotient ). For example: 

10 div  2 = 5
22 div  7 = 3
65 div 10 = 6
---
N div K
The result of N div K shows how many times K «fits inside» N.

Examples:
Example 1. 

How many Kilobytes are in x bytes?
Answer: x div 1024

Example 2.

x is a time in seconds
How many seconds have passed since the last minute?
Answer: x mod 60

Mod

The mod operation calculates the remainder after dividing two integers. In this case, if the numbers are evenly divisible, the remainder is zero. Example:

10 mod  2 = 0
22 mod  7 = 1
65 mod 10 = 5
---
N mod K
The result of N mod K shows the «remainder of N» after the maximum number of K is «thrown» out of N.

Examples
Example 3.

x mod 2 = 0 ⟹ x – even number
x mod 2 <> 0 ⟹ x – odd number

Example 4.

var x := 1234;
var LastDigit := x mod 10; // 4
var NumWithoutLastDigit := x div 10; // 123

Example 5.

// x is a 3-digit number. What is the second digit?
// Answer:
var x := ReadInteger('Enter x:'); // 456
x := x div 10; // 45
Print(x mod 10); // 5

Bitwise number operation

Formula to get the number out of the quotient, divisor and remainder

By parsing any integer N into two components — the quotient d and remainder m, using the same divisor K and operations div and mod, you will then easily restore this number by the formula:

If we have 10:
10 div 2 = 5
10 mod 2 = 0
       ↓
5 * 2 + 0 = 10


Examples:

10 div  2 = 5;  10 mod  2 = 0    =>   5*2 + 0  = 10
22 div  7 = 3;  22 mod  7 = 1    =>   3*7 + 1  = 22
65 div 10 = 6;  65 mod 10 = 5    =>   6*10 + 5 = 65

Standard Form Of A Number

Any number can be disassembled into digits using powers of 10 and specified operations. This rule is called the standard form of a number:

123 = 1*100 + 2*10 + 3

Therefore, to make a certain number a hundred, you need to multiply it by 100 (as we have for digit 1 we have 1 * 100 = 100).

3 Rules To Get Digits Of Three-Digit Integer (three bit width)
    It can be seen that:

  • to get hundred (first digit) of three-digit number, you need to calculate the quotient after dividing that number by 100 (number div 100, e.g. 123 div 100 = 1)
  • to get ten (second digit), you need to calculate the remainder after dividing that number by 100, and then — the quotient of dividing the result by 10 (1. number mod 100: 123 mod 100 = 23; 2. 23 div 10 = 2) (also there is another way)
  • to get unit number (third digit), you need to calculate remainder after dividing this number by 10 (number mod 10: 123 mod 10 = 3).
    For numbers with a different bit width, these algorithms can be changed.