Lesson #2. Working with digits of a number

Theory

Lecture # 2 in pdf format

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

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

// 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 

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

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).

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.

Labs and Tasks

To complete the tasks, please, follow the rules.

Please enter the distance in centimeters
>>> 245
The distance in meters is  2.45 

Please enter the mass in kilos
>>> 4527
The mass in tons is 4 

Please enter the age in months
>>> 37
The age in years is 3 

Please enter a two digit number
>>> 37
The first and second are  7,  3 
+++
Please enter a two digit number:
>>> -35
The first and second are 5, 3
+++
Please enter a two digit number:
>>> 90
The first and second are 0, 9 

please enter a two digit number
>>> 73
you entered 73. The result is 10, 21 

Please enter a three-digit number
>>> -105
The three digits are : 1, 0, 5 

Please enter a three-digit number
>>> -745
You entered: 745. The addition of its three digits is equal to 16 

Please enter two digits
>>> 3
>>> 5
The new number is 35 

3, 5 => 35
7, 0 => 70
0, 4 => 4

please enter a two digit number
>>> -57
you entered -57. The result is -75

35 =>  53
-10 => -1

please enter a three digit integer
>>> 846
After swaping the left and middle digits, the result is 486 

please enter a three digit integer
>>> -734
After making a cyclic shift of digits to the left, the result is -347 

 123 =>  231
-602 => -26

please enter a three digit integer
>>> -184
After a cyclic shift of digits to the right, the result is -418