GCSE Maths Practice: factors-and-multiples
This Higher-level question focuses on finding the greatest common divisor (GCD) and links the concept to simplifying fractions and ratio reasoning.
Always test divisibility first. If one number divides the other, the smaller is the GCD. Otherwise, use prime factors or Euclid’s method.
GCD and Fraction Simplification — Higher Concept Link
The greatest common divisor (GCD) or highest common factor (HCF) is the largest whole number that divides two or more integers exactly. At Higher GCSE level, this concept connects directly to simplifying fractions, ratio reduction, and even algebraic factorisation. Understanding how to find the GCD efficiently ensures accuracy when working with both numbers and algebraic expressions.
Using Prime Factorisation
Each integer can be written uniquely as a product of primes. To find the GCD, compare the prime factorizations and multiply the primes that appear in both numbers, using their lowest powers. This approach avoids missing hidden factors.
Example (different numbers): Find the GCD of 210 and 126.
210 = 2 × 3 × 5 × 7
126 = 2 × 3^2 × 7
Common primes: 2 × 3 × 7 = 42
Therefore, GCD = 42.
Connecting GCD to Fraction Simplification
In GCSE problems, fractions are simplified by dividing both numerator and denominator by their GCD. For example:
\(\frac{196}{294} = \frac{196 \div 98}{294 \div 98} = \frac{2}{3}.\)This shows how the GCD links directly to ratio and fraction work. Using prime factorisation ensures the simplification is complete and avoids partial reductions.
Alternative Approach — Euclidean Algorithm
For larger or unfamiliar numbers, the Euclidean algorithm provides a faster, division-based method.
Find GCD(882, 98):
882 ÷ 98 = 9 remainder 0 → GCD = 98.
This demonstrates that sometimes one number is already a multiple of the other, making the GCD immediately visible.
Common Errors
- Assuming the smaller number is always the GCD (only true if it divides the larger exactly).
- Using the highest powers instead of the lowest in prime factorisation.
- Stopping simplification too early when working with fractions or ratios.
Real-Life Applications
The GCD helps in reducing ratios, scaling models, comparing probabilities, and even finding the largest tile size that evenly fits dimensions in geometry. For instance, if a rectangle measures 98 cm by 196 cm, the largest square tile that fits evenly on both sides has a side length of 98 cm—the GCD of the dimensions.
FAQ
Q: Can the GCD ever equal one of the numbers?
A: Yes, if the smaller divides the larger exactly—as in 98 and 196, where 98 × 2 = 196.
Q: Why is the GCD useful for fractions?
A: It provides the simplest possible equivalent fraction, ensuring numerator and denominator share no common factors.
Q: Is the GCD of two even numbers always even?
A: Yes, because both include 2 as a factor.
Study Tip
When you see large numbers that look like doubles or triples of each other, first check for divisibility rather than listing or factorising. Spotting that 196 = 2 × 98 instantly gives the GCD without further work.
Summary
Understanding GCDs bridges arithmetic and algebra. Whether simplifying fractions, comparing ratios, or recognising common factors in algebraic terms, mastering both the prime factorisation and Euclidean approaches provides a strong foundation for Higher GCSE Maths.