Divisibility, Factors & Multiples • Topic 3 of 5

GCD (HCF)

The greatest common divisor GCD(a, b) — also called the highest common factor, HCF — is the largest integer that divides both a and b. The reliable method is prime factorisation: take each shared prime raised to its lowest power. For 48 = 2⁴·3 and 60 = 2²·3·5, the common primes are 2 (lowest power 2²) and 3, so GCD = 4 × 3 = 12. GCD answers "what is the largest equal group I can make?" — the biggest identical bundles, the largest square tile that fits a rectangle exactly, or the largest number of gift bags with the same contents. If two numbers share no prime factors, their GCD is 1 and they are called coprime (or relatively prime), which is exactly the condition for a fraction to be fully reduced.

✅ Solved examples

1. Find GCD(48, 60).
48 = 2⁴·3 and 60 = 2²·3·5. Take shared primes at the lowest power: 2² and 3¹. GCD = 4 × 3 = 12.
2. A florist has 24 roses and 36 tulips and wants identical bunches using all flowers. What is the greatest number of bunches?
The greatest equal split is GCD(24, 36). 24 = 2³·3, 36 = 2²·3². GCD = 2²·3 = 12. So 12 bunches (each with 2 roses and 3 tulips).
3. Reduce 84/126 to lowest terms.
GCD(84, 126): 84 = 2²·3·7, 126 = 2·3²·7. Common at lowest powers: 2·3·7 = 42. So 84/126 = 2/3.
4. Quantitative Comparison. Quantity A: GCD(15, 28). Quantity B: 1. Which is greater?
15 = 3·5 and 28 = 2²·7 share no prime factors, so GCD = 1. They are coprime. Quantity A = Quantity B = 1; equal.

✏️ Practice — try these, take hints as needed

1. Find GCD(18, 24).
18 = 2·3², 24 = 2³·3.
Lowest shared powers.
2 × 3.
6
2. Find GCD(35, 49).
35 = 5·7, 49 = 7².
Shared prime is 7.
Lowest power 7¹.
7
3. Largest square tile (whole cm) that exactly tiles a 60 cm by 48 cm floor?
Tile side = GCD.
GCD(60, 48).
60 = 2²·3·5, 48 = 2⁴·3.
12 cm
4. Reduce 45/72 to lowest terms.
GCD(45, 72).
45 = 3²·5, 72 = 2³·3².
Common 3² = 9.
5/8
5. Are 16 and 27 coprime?
16 = 2⁴.
27 = 3³.
Shared factors?
Yes (GCD = 1)

📝 Topic test — 8 questions

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