Important Formulae: Numbers | Mathematics & Pedagogy Paper 1 for CTET & TET Exams - CTET & State TET PDF Download

What is Number System?

A number system, or system of numeration, is a method of representing numbers or symbols in a consistent way. In simple terms, it refers to the way numbers are written.

Here is a list of Important formulas that candidates need to master to crack the exam successfully.

Important Formulas in Number System

1. Sum of the first n natural numbers
   Formula:
   Example: Find the sum of the first 5 natural numbers.
   Solution: Using the formula, n = 5,
2. Sum of the squares of the first n natural numbers
   Formula:Example: Find the sum of the squares of the first 4 natural numbers.
   Solution: Using the formula, n = 4,
3. Sum of the cubes of the first n natural numbers
   Formula:Example: Find the sum of the cubes of the first 3 natural numbers.
   Solution: Using the formula, n = 3,
   
4. Sum of the first n odd numbers
   Formula: Sum of first n odd numbers = n²
   Example: Find the sum of the first 5 odd numbers.
   Solution: Using the formula, n = 5,
   Sum of first 5 odd numbers = 5² = 25
5. Sum of the first n even numbers
   Formula: Sum of first n even numbers = n × (n + 1)
   Example: Find the sum of the first 4 even numbers.
   Solution: Using the formula, n = 4,
   Sum of first 4 even numbers = 4 × (4 + 1) = 4 × 5 = 20

Important Formulas in Algebra

• (a + b)(a - b) = (a² - b²)
• (a + b)² = (a² + b² + 2ab)
• (a - b)² = (a² + b² - 2ab)
• (a + b + c)² = a² + b² + c² + 2(ab + bc + ca)
• (a³ + b³) = (a + b)(a² - ab + b²)
• (a³ - b³) = (a - b)(a² + ab + b²)
• (a³ + b³ + c³ - 3abc) = (a + b + c)(a² + b² + c² - ab - bc - ac)
• When a + b + c = 0, then a³ + b³ + c³ = 3abc
• (a + b)ⁿ = aⁿ + (ⁿC₁)a^n-1b + (ⁿC₂)a^n-2b² + … + (ⁿC_n-1)ab^n-1 + bⁿ

Divisibility Rules

• Divisibility by 2 → A number that is even or a number whose last digit is an even number i.e. 0, 2, 4, 6, and 8.
• Divisibility by 3 → The sum of all the digits of the number should be divisible by 3.
• Divisibility by 4 → Number formed by the last two digits of the number should be divisible by 4 or should be 00.
• Divisibility by 5 → Numbers having 0 or 5 as their ones place digit.
• Divisibility by 6 → A number that is divisible by both 2 and 3.
• Divisibility by 7 → Subtracting twice the last digit of the number from the remaining digits gives a multiple of 7.
• Divisibility by 8 → Number formed by the last three digits of the number should be divisible by 8 or should be 000.
• Divisibility by 9 → The sum of all the digits of the number should be divisible by 9.
• Divisibility by 10 → Divisibility rule for 10 states that any number whose last digit is 0, is divisible by 10.
• Divisibility by 11 → The difference of the sums of the alternative digits of a number is divisible by 11.
• Divisibility by 12 → A number that is divisible by both 3 and 4.
• Divisibility by 13 → For any given number, to check if it is divisible by 13, we have to add four times of the last digit of the number to the remaining number and repeat the process until you get a two-digit number.  Now check if that two-digit number is divisible by 13 or not. If it is divisible, then the given number is divisible by 13.
• Divisibility by 14→ A number divisible by both 2 and 7.
• Divisibility by 16 → Last four-digit divisible by 16
• Divisibility by 27 → Sum of blocks of 3 (taken a right to left) divisible by 27