KS3 · Maths

Prime numbers and prime factor decomposition

Split 36 into any two numbers that multiply to make it, then keep splitting every piece you can. Try a different start. You always finish with the same four numbers. Why?

Maths · Prime factorisation

Different starts, same finish

Choose any first split of 36, then keep choosing until only primes are left. Then step back and try a completely different start.

First split → Second split → Final split

  • 36 has factor pairs besides 1 × 36, so it isn't prime. Choose a pair to start with.

6 routes through 36.

On 36. 4 branches to choose from.

A prime is a number whose only factor pair is 1 and itself, like 2, 3 or 7, so a prime can't be split any further. 36 is not prime, so it can be split. Choose how.

Watch out: The number of routes is only how many ways there are to choose your splits. It isn't a factor of 36, and it doesn't change the finish. Every route ends on the same primes.

Maths · Primes

Prime, composite, or neither?

Sort each number by its factors: prime, composite or neither. Then switch to the sieve sort.

How many factors does it have?

Still to sort

Prime (0)

Exactly two factors: 1 and itself

Where the line is: Prime means exactly two factors. A number with a factor pair other than 1 and itself has more than two.

Composite (0)

More than two factors

Where the line is: If you can find a factor pair that isn't 1 and the number itself, it is composite.

Neither (0)

Only one factor

8 of 8 still to sort.

Check a number's factor pairs before you decide. Then re-sort the same numbers using the Sieve of Eratosthenes and see if the two methods agree.

Watch out: 1 is the awkward one. Count its factors before you place it.

Writing a number as a product of primes

Problem

Write 180 as a product of its prime factors, using index notation.

Maths · Using a known product

Don't start again. Reuse it.

You know 180 = 2² × 3² × 5. Use it to write 1,800 as a product of primes without starting from scratch.

  1. We know 180 = 2² × 3² × 5.
  2. 1,800 = 180 × 10, so keep the primes of 180 and add the primes of 10.
  3. missing step
Which line is step 3?

Maths · Reading a product of primes

Facts you can read without multiplying

Don't work out any of these numbers. Look at the primes and the indices, then tick every property that is true.

2² × 3 × 11²
2⁴ × 3³ × 5 × 7²
3² × 5² × 7²
2² × 5²
2 × 3² × 7

Maths · Common factors

What do two numbers share?

60 = 2 × 2 × 3 × 5 and 24 = 2 × 2 × 2 × 3. Tick which product (or both, or neither) each item appears in. Treat every repeated 2 as its own item.

  • A Prime factors of 60
  • B Prime factors of 24
  1. 2 (first)
  2. 2 (second)
  3. 3
  4. 5
  5. 2 (third)
  6. 1

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Break a number down until nothing more can be split, and see why every route ends in the same place.

What you need to know

  • A prime number is an integer greater than 1 with exactly two factors: 1 and itself. 1 is not prime, because it has only one factor.
  • A composite number has more than two factors. Every integer greater than 1 is either prime or composite.
  • To write a number as a product of primes, start with any factor pair, keep splitting every factor that isn't prime, then write the primes in numerical order.
  • Each composite number has exactly one product of primes. A different starting pair gives the same primes, and only the order can change. It is a multiplication, never an addition.
  • Repeated primes can be tidied with index notation, like 2 × 2 × 3 × 5 × 5 = 2² × 3 × 5². An index of 1 is not written.
  • You can read facts from a product of primes without evaluating it: a multiple of 10 needs a 2 and a 5, and a square number has every index even.

The big picture

Every integer greater than 1 is either prime (exactly two factors) or composite (more than two). Every composite number can be written as one product of primes, however you start splitting it, and that product lets you read facts about the number without working it out.

Key points

1Prime: exactly two factors. Composite: more than two. 1 is neither, because it has only one factor.
22 is the only even prime, and 5 is the only prime ending in 5. Any number ending in zero is a multiple of 10, so it is never prime.
3Prime factors are the factors of a number that are themselves prime.
4Different starting factor pairs lead to the same primes. The product of primes is unique apart from order.
5A known product of primes lets you write related numbers without starting again, and evaluating a product of primes gives back one whole number.
6Common factors of two numbers come from the primes their products share, and 1 must be added to the list because it is never in a product of primes.

Worked example

Problem

Is 84 a multiple of 10? Decide using its prime factors, not a division.

⚠ Watch out

Calling 1 a prime number. It has only one factor, and a prime needs exactly two, so 1 is neither prime nor composite. A second slip is adding the primes instead of multiplying them: 2 + 3 + 5 is not 30, but 2 × 3 × 5 is.

🧠

Memory hook

Primes are the unbreakable bricks that numbers are built from. Take a number apart however you like and you always end up with the same bricks. (Unlike real bricks, a prime can be used more than once, like the two 2s in 2 × 2 × 3.)

✓

Check yourself

Take 48. Split it as 6 × 8, then again as 4 × 12, splitting until every factor is prime. Do both finish with the same primes? Write them with indices.

Flashcards

(14)
What is a prime number?
An integer greater than 1 with exactly two factors: 1 and itself. 7 is prime because its only factor pair is 1 and 7.
Why is 1 not a prime number?
It has only one factor, and a prime number has exactly two.
What is a composite number?
An integer with more than two factors. Every integer greater than 1 is either prime or composite.
How do you test whether 27 is prime using factor pairs?
List its factor pairs: 1 × 27 and 3 × 9. A second pair means more than two factors, so 27 is not prime.
What is the Sieve of Eratosthenes?
A way to find primes on a 100-square. Cross out 1, then the multiples of 2, 3, 5 and 7 (but not those primes themselves). The numbers left standing are prime.
Which primes are special: the even one and the one ending in 5?
2 is the only even prime, because all other even numbers are multiples of 2. 5 is the only prime ending in 5. A number ending in zero is a multiple of 10, so it is never prime.
What are prime factors?
The factors of a number that are themselves prime. Every composite number has them.
What does it mean that the product of primes is unique?
There is only one way to write a composite number as primes. Starting from a different factor pair gives the same primes, and only the order can change.
How do you write a number as a product of its primes?
Start with any factor pair. Keep splitting every factor that isn't prime. Then write the primes in numerical order as a multiplication.
How does index notation shorten 2 × 2 × 3 × 5 × 5?
Write each repeated prime once, with an index counting the repeats: 2² × 3 × 5². An index of 1 is not written.
If 360 = 2³ × 3² × 5, how do you write 720 without starting again?
720 = 360 × 2, so add one more 2: 720 = 2⁴ × 3² × 5.
What must a multiple of 10 have in its product of primes?
A factor of 2 and a factor of 5.
How can you spot a square number from its product of primes?
Every index is even. A square number is the product of two identical integers, so an odd index means it can't be one.
Why must you write 1 in a list of common factors by hand?
1 is not prime, so it never appears in a product of primes. It is still a common factor of any two numbers.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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