Integers | Year 8 Mathematics (Cambridge) PDF Download

Factors, Multiples, and Primes

Factors

• Factors are numbers that divide another number exactly without leaving a remainder. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12.
• Examples:
  • Factors of 20: 1, 2, 4, 5, 10, 20
  • Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Multiples

• Multiples of a number are obtained by multiplying that number by an integer. For example, the multiples of 5 are 5, 10, 15, 20, and so on.
• Examples:
  • First five multiples of 3: 3, 6, 9, 12, 15
  • First five multiples of 7: 7, 14, 21, 28, 35

Primes

• Prime numbers are numbers greater than 1 that have no factors other than 1 and themselves. For example, 2, 3, 5, 7, and 11 are prime numbers.
• Examples:
  • Prime numbers between 1 and 20: 2, 3, 5, 7, 11, 13, 17, 19
  • Prime numbers between 20 and 40: 23, 29, 31, 37

Multiplying and Dividing Integers

Multiplying Integers

• When multiplying integers, the product of two positive integers is positive, and the product of two negative integers is also positive. The product of a positive integer and a negative integer is negative.
• Examples:
  • 3 × 4 = 12
  • (−3) × (−4) = 12
  • 3 × (−4) = −12

Dividing Integers

• When dividing integers, the quotient of two positive integers is positive, and the quotient of two negative integers is also positive. The quotient of a positive integer and a negative integer is negative.
• Examples:
  • 12 ÷ 4 = 3
  • (−12) ÷ (−4) = 3
  • 12 ÷ (−4) = −3

Square Roots and Cube Roots

Square Roots

• The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 16 is 4 because 4 × 4 = 16.
• Examples: 
  • √25 = 5
  • √49 = 7

Cube Roots

• The cube root of a number is a value that, when multiplied by itself three times, gives the original number. For example, the cube root of 27 is 3 because 3 × 3 × 3 = 27.
• Examples:
  • ∛8 = 2
  • ∛64 = 4

Indices

Basic Indices

Indices, also known as exponents or powers, show how many times a number is multiplied by itself. For example, 2³ means 2 x 2 x 2 = 8.

Examples:

3² = 3 × 3 = 9

5³ = 5 × 5 × 5 = 125

Laws of Indices

There are several laws of indices that make calculations easier:

• Multiplication: a^m × aⁿ = a^m + n
  • Example: 2³ × 2² = 2^{3 + 2} = 2⁵ = 32 
• Division: a^m ÷ aⁿ = a^m−n 
  • Example: 2⁵ ÷ 2² = 2⁵⁻² = 2³ = 8 
• Power of a Power: (a^m)ⁿ = a^mn 
  • Example: (2³)² = 2^3×2 = 2⁶ =64 
• Zero Index: a⁰ = 1 for any 𝑎 ≠ 0 
  • Example: 5⁰ = 1 
• Negative Index:
  • Example: