Revision Notes: Playing with Numbers | Mathematics Class 6 ICSE PDF Download

Factors and Multiples 

• A factor of a number is an exact divisor of that number. 6 = 1 x 6 = 2 x 3: factors of 6 are 1, 2, 3 and 6 
• Every number is a factor of itself, and 1 is a factor of every number
• Every factor of a number is less than or equal to the given number 
• Every number is a multiple of each of its factors. Factors of 12 are 1, 2, 3, 4, 6 and 12  :  3 x 4 = 12,  2 x 6 = 12, 12  x 1 = 12 
• Every multiple of a given number is greater than or equal to that number. 
• Every number is a multiple of itself 
• Factors of a given number are finite, while its multiples are infinite
• If two given numbers are divisible by a number, then their sum (or difference)  is also divisible by that number: 6 is divisible by 3, and 18 is divisible by 3. Hence 24 (= 18 + 6) and 12 (= 18 - 6) are also divisible by 3 
• A number for which the sum of all its factors is equal to twice the number is called a perfect number: 
  Ex.  6 and 28

Prime and Composite Numbers 

• A number having only 2 factors,  namely 1 and the number itself, is a prime number: 2, 3, 5, 7, 11, etc.
• Numbers that have more than two factors are called composite numbers: 4, 6, 8, 9, 10, 12, etc.
• Number 1 is neither prime nor composite. 
• The number 2 is the smallest prime number and is even. Every prime number other than 2 is odd.
• Two numbers with only 1 as a common factor are called co-prime numbers: 15 and 16,  8 and 9, 25 and 3,3, etc.
• If a number is divisible by another number, then it is divisible by each of the factors of that number. 24 is divisible by 8: it is also divisible by 2 and 4, which are factors of 8.
• A number divisible by two co-prime numbers is also divisible by their product. 8 and 9 are co-primes, and 8 x 9 = 72 
  144 is divisible 9 : 9 x 16 = 144 ,   144 is divisible by 8 : 8 x 18 = 144 
  144 is also divisible by 72 (product of 8 and 9) : 72 x 2 = 144

Test of divisibility

Prime factorization

The prime factorisation of 36 is = 2 × 2 × 3 × 3. i.e. the only prime factorisation of 36.

The Highest Common Factor(HCF)

• The HCF of two or more given numbers is the highest (or greatest) of their common factors. 
• It is also known as the Greatest Common Divisor (GCD)

20 = 2 x 2 x 5 
28 = 2 x 2 x 7 
The common factors of 20 and 28 are 2, 2 
Hence HCF (20, 28) = 2 x 2 = 4

Steps to Find the HCF of Three or More Numbers

• Find the HCF of any two numbers among the given numbers 
• Find the HCF of the third number and the HCF obtained in the previous step 
• The last HCF will be the HCF of all given numbers.

The Lowest Common Multiple (LCM)

• Multiple: When a number divides another number completely, the second number is called a multiple of the first number 
• Common Multiple: A number which can be divided completely by the given two or more numbers is called their common multiple. 
• For any two coprime numbers, their LCM is equal to their product.

Index Method 

• 12 = 2×2×2×3 = 2³ × 3; 
• 36 = 2×2×3×3 = 2² × 3²; 
• 40 = 2×2×2×5 = 2³ × 5 
• LCM = 2³ × 3² × 5 = 360 

Division Method

• In this prime factorisation, the maximum number of times the prime factor 2 occurs is three
• Similarly, the maximum number of times the prime factor 3 occurs is two
• The prime factor 5 occurs only once
• Thus, LCM = (2 × 2 × 2) × (3 × 3) × 5 = 360

For any two numbers, the product of their LCM and HCF is equal to product of the numbers.