Greatest Common Divisor
Largest number dividing two numbers exactly; found with prime factors or Euclid.
Key idea
Largest number dividing two numbers exactly; found with prime factors or Euclid.
In depth
The GCD is the largest number dividing two numbers exactly. Use it to simplify fractions to lowest terms. Find it from prime factorizations by taking the shared primes at their smallest powers.
Worked example
GCD(36, 48): 36 = 2²×3², 48 = 2⁴×3. Shared: 2²×3 = 12.
Formula
Real-world uses
- Cutting a rectangle into the largest identical square tiles
- Simplifying ratios in recipes
Common mistake
Taking the largest power instead of the smallest; that gives the LCM.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Greatest Common Divisor?
Largest number dividing two numbers exactly; found with prime factors or Euclid. It is a topic at the intermediate level in Arithmetic & Number Sense.
What is the formula for Greatest Common Divisor?
The key formula is gcd(12, 18) = 6.
Where is Greatest Common Divisor used in real life?
Real-world examples: Cutting a rectangle into the largest identical square tiles; Simplifying ratios in recipes.
What should I learn before Greatest Common Divisor?
Make sure you are comfortable with Prime Factorization, the previous topic in Arithmetic & Number Sense.
What is a common mistake with Greatest Common Divisor?
Taking the largest power instead of the smallest; that gives the LCM.
How can I practise Greatest Common Divisor?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Greatest Common Divisor?
Move on to Least Common Multiple next.
Related intermediate topics in Arithmetic & Number Sense
- Long Multiplication – Multiply by each digit of the multiplier, shift by place value, then add partial products.
- Long Division – Divide, multiply, subtract and bring down repeatedly to get the quotient.
- Factors and Multiples – Factors divide a number exactly; multiples are its products with whole numbers.
- Divisibility Rules – Quick tests: by 2 last digit even, by 3 digit-sum divisible by 3, by 5 ends 0/5, by 9 digit-sum divisible by 9.
- Prime Numbers – A prime has exactly two positive divisors: 1 and itself.
- Composite Numbers – Numbers greater than 1 with more than two factors.