Factors, multiples, and primes (age 11+)
Use the concepts and vocabulary of prime numbers, factors, multiples, common factors, common multiples, highest common factor (HCF), lowest common multiple (LCM), and prime factorisation including product notation and the unique factorisation property
What a learner can do afterwards
- Express any integer as a product of its prime factors using index notation
- Find the HCF and LCM of two numbers using prime factorisation
- Apply the unique factorisation theorem to explain why every number has exactly one set of prime factors
1 · Read
Say you have 24 strawberries and 36 blueberries, and you want to arrange them into the biggest matching trays with none left over. You already know what prime numbers, factors, and multiples are. Now you'll break any whole number down into the primes that build it, and write that breakdown quickly using index notation.
Take 24 = 2³ × 3 and 36 = 2² × 3². For the HCF, multiply the primes they share, using the lower power each time: 2² × 3 = 12. For the LCM, multiply every prime that appears, using the higher power each time: 2³ × 3² = 72. So the HCF of 24 and 36 is 12, and the LCM is 72.
Every whole number greater than 1 has exactly one set of prime factors. Divide by 2 first or by 3 first, and you still land on the same primes in the end. That's why it's called unique factorisation.
Break a number into primes with index notation, then use those same primes to find the HCF (shared primes, lower power) and LCM (all primes, higher power).
More to read
Prime Factorization and the Least Common Multiple
Every whole number bigger than 1 is either prime, or it can be broken down into prime numbers multiplied together. That breakdown is called the prime factorization. To find it, keep dividing by the smallest prime that fits until only primes are left, then write repeated primes using exponents.
The lowest common multiple (LCM) of two numbers is the smallest number that both of them divide into evenly. One good way to find it is to write the prime factorization of each number, then take the highest power of every prime that shows up, and multiply those together.
It helps to know your divisibility tests before you start, like checking whether a number ends in 0 or 5 to see if 5 divides into it evenly.
Find Multiples and Factors
A multiple of a number is what you get when you multiply it by a counting number, like 3, 6, 9, 12 for the number 3. A factor of a number is a number that divides evenly into it with nothing left over.
Divisibility tests are quick checks that tell you if a number divides evenly into another, without doing the whole division. For example, a number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.
To find all the factors of a number, try dividing it by 1, then 2, then 3, and so on, keeping every pair that works. A prime number has exactly two factors: 1 and itself. A composite number has more than two factors.
Greatest Common Factor and Factor by Grouping
Before factoring bigger expressions, it helps to practice breaking plain numbers into primes. For example, 56 splits into 2 × 2 × 2 × 7, or 2³ × 7 in index notation.
The greatest common factor (GCF) of two numbers is the largest number that divides evenly into both of them. You can find it by writing the prime factorization of each number, then multiplying together the primes they share, using the lowest power of each.
The least common multiple (LCM) works the other way. It is the smallest number that both original numbers divide into. For 18 and 24, you would compare their prime factorizations and take the highest power of each prime that appears.
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Where it sits
18 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.