Factors and Multiples • Topic 4 of 7

Greatest Common Factor (GCF)

The greatest common factor (GCF, also HCF) of two numbers is the largest number that divides both. The cleanest method uses prime factorization: take each shared prime raised to its lowest power. For example 48 = 2⁴ × 3 and 60 = 2² × 3 × 5 share 2² × 3 = 12, so GCF = 12. The Euclidean algorithm (repeated remainders) is faster for large numbers. GCF answers “largest equal groups / biggest tile” word problems, and two numbers whose GCF is 1 are called coprime. Remember the identity GCF × LCM = product of the two numbers.

✅ Solved examples

1. Find the GCF of 24 and 36.
Factor: 24 = 2³ × 3, 36 = 2² × 3². Take the lowest power of each shared prime: 2² × 3¹ = 4 × 3 = 12. So GCF = 12.
2. A florist has 30 roses and 45 tulips and wants identical bouquets using all flowers. What is the greatest number of bouquets?
The number of bouquets must divide both 30 and 45, so take the GCF. 30 = 2 × 3 × 5, 45 = 3² × 5; shared = 3 × 5 = 15. Greatest number of bouquets = 15.
3. Are 14 and 25 coprime?
14 = 2 × 7 and 25 = 5². They share no prime factor, so their GCF is 1 — yes, 14 and 25 are coprime.
4. The GCF of two numbers is 6 and their LCM is 72. If one number is 24, find the other.
Use GCF × LCM = product: 6 × 72 = 432. The other number = 432 ÷ 24 = 18.

✏️ Practice — try these, take hints as needed

1. Find the GCF of 18 and 48.
Write each as a product of primes.
18 = 2 × 3², 48 = 2⁴ × 3.
Take the lowest power of each shared prime: 2¹ × 3¹.
6.
2. Two ribbons of length 36 cm and 60 cm are cut into equal pieces with none left over. What is the greatest possible piece length?
“Greatest equal pieces” signals GCF.
36 = 2² × 3², 60 = 2² × 3 × 5.
Shared lowest powers: 2² × 3.
12 cm.
3. Find the GCF of 17 and 51.
Check whether the smaller divides the larger.
51 ÷ 17 = 3 exactly.
If one divides the other, the GCF is the smaller number.
17.
4. The GCF of two numbers is 8 and one number is 8. What can the other number be (give the smallest above 8)?
8 must divide the other number.
The other = 8k, and GCF(8, 8k) = 8 needs k to share no extra factor of… actually any multiple of 8 works for GCF 8 as long as it is a multiple of 8.
Smallest multiple of 8 greater than 8 is 16.
16.
5. GCF × LCM of two numbers is 600 and their GCF is 10. What is their LCM?
Use GCF × LCM = product of the numbers — here you are given GCF × LCM directly as 600? Re-read: product = 600.
Then LCM = product ÷ GCF.
LCM = 600 ÷ 10.
60.

📝 Topic test — 8 questions

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