MP Board Class 6 Mathematics

Quick Revision Notes & Formula Sheet

Chapter 3: Playing with Numbers (संख्याओं के साथ खेलना)

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1. Factors and Multiples (गुणनखंड और गुणज)

• Factor (गुणनखंड): A factor of a number is an exact divisor of that number.

  • Example: Factors of 12 are 1, 2, 3, 4, 6, 12.

• Properties of Factors:

  • 1 is a factor of every number.
  • Every number is a factor of itself.
  • Every factor of a number is less than or equal to that number.
  • The number of factors of a given number is finite (countable).

• Multiple (गुणज): A multiple of a number is obtained by multiplying it by any natural number.

  • Example: Multiples of 5 are 5, 10, 15, 20, 25, ...

• Properties of Multiples:

  • Every multiple of a number is greater than or equal to that number.
  • The number of multiples of a given number is infinite (uncountable).
  • Every number is a multiple of itself.

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2. Classification of Numbers

• Prime Numbers (अभाज्य संख्याएं): Numbers having exactly two factors — 1 and the number itself.
  • Examples: 2, 3, 5, 7, 11, 13, 17, 19, ...
  • 2 is the smallest and the only even prime number.
• Composite Numbers (भाज्य संख्याएं): Numbers having more than two factors.
  • Examples: 4, 6, 8, 9, 10, 12, ...
• Special Note on 1:
  • 1 is neither prime nor composite (it has only one factor: itself).
• Even Numbers (सम संख्याएं): Numbers that are multiples of 2 (ends with 0, 2, 4, 6, 8).
• Odd Numbers (विषम संख्याएं): Numbers that are not multiples of 2 (ends with 1, 3, 5, 7, 9).
• Co-prime Numbers (सह-अभाज्य संख्याएं): Two numbers having only 1 as a common factor.
  • Example: (4, 9), (8, 15)
  • Two consecutive natural numbers are always co-prime.
• Twin Primes (जुड़वां अभाज्य): Two prime numbers whose difference is 2.
  • Examples: (3, 5), (5, 7), (11, 13), (17, 19)

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3. Divisibility Rules (विभाज्यता के नियम)

Divisible by	Test / Condition	Example
2	Last digit is 0, 2, 4, 6, or 8	248 (ends in 8)
3	Sum of digits is divisible by 3	381 (3+8+1 = 12, divisible by 3)
4	Number formed by last two digits is divisible by 4	512 (12 is divisible by 4)
5	Last digit is 0 or 5	105, 240
6	Number is divisible by both 2 and 3	432 (Even and sum 4+3+2=9)
8	Number formed by last three digits is divisible by 8	4160 (160 is divisible by 8)
9	Sum of digits is divisible by 9	729 (7+2+9 = 18, divisible by 9)
10	Last digit is 0	500
11	Difference between sum of digits at odd places and even places is 0 or divisible by 11	1331 ((1+3) - (3+1) = 0)

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4. Prime Factorization (अभाज्य गुणनखंडन)

When a number is expressed as a product of prime numbers, it is called Prime Factorization.

• Example: Prime factorization of 36
  • 36 = 2 * 2 * 3 * 3

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5. HCF and LCM

• HCF (Highest Common Factor / महत्तम समापवर्तक - म.स.):

  • The largest number that divides two or more numbers without leaving a remainder.
  • Also known as GCD (Greatest Common Divisor).
  • Methods to find HCF: Prime Factorization Method, Division Method.

• LCM (Lowest Common Multiple / लघुत्तम समापवर्त्य - ल.स.):

  • The smallest positive number that is a multiple of two or more numbers.
  • Methods to find LCM: Prime Factorization Method, Common Division Method.

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6. Key Formulas & Important Rules

Formula 1: Product Relation (For two numbers)

Product of two numbers = HCF * LCM

First Number * Second Number = HCF * LCM

Derived Formulas:

HCF = (First Number * Second Number) / LCM

LCM = (First Number * Second Number) / HCF

Important Properties:

1. For Co-prime Numbers:

   • HCF of two co-prime numbers = 1
   • LCM of two co-prime numbers = Product of the two numbers

2. General Rules:

   • HCF of given numbers is always less than or equal to the smallest number.
   • LCM of given numbers is always greater than or equal to the largest number.
   • HCF of given numbers is always a factor of their LCM.

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Quick Memory Check Tips:

• Smallest Prime Number: 2
• Smallest Composite Number: 4
• Smallest Odd Prime Number: 3
• Only Even Prime Number: 2