Year 7 · Unit 1 Number systems and place value · Lesson 4 of 7

Does it matter how you split a number?

Split 60 as 6 × 10 or as 4 × 15 and keep splitting. The branches look different, but the primes at the bottom are always the same.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Write 2 × 2 × 2 × 5 as a power times a number.
    Answer2³ × 5, which is 8 × 5 = 40.
  2. Is 51 prime? Explain.
    AnswerNo: 51 = 3 × 17. The digit sum 6 is a multiple of 3, so 3 is a factor.
  3. List the factor pairs of 20.
    Answer1 × 20, 2 × 10 and 4 × 5.
Words used in this lesson
prime factor
a factor that is a prime number. For example: the prime factors of 12 are 2 and 3.
factor tree
a diagram that splits a number into factors. For example: 60 splits into 6 and 10.
product
the result of multiplying. For example: the product of 3 and 5 is 15.
index form
a number written with powers of primes. For example: 60 = 2² × 3 × 5.
unique
only one way, apart from the order. For example: the prime factorization is unique.
perfect square
a number that is a whole number squared. For example: 36 = 6².

Splitting until only primes are left

A coach with 60 players wants equal teams, and needs to know every team size that works. Every whole number above 1 is built from primes the way every word is built from letters, and the primes of 60 show all the team sizes at once. To find the primes, split the number into a pair of factors, split any factor that is not prime, and keep going. A picture of this is a factor tree. A thick coloured ring on a number means it is prime and stops.

Predict first: 60 is 6 × 10, and also 4 × 15. If two students split it differently and keep going until everything is prime, will they find different primes? Grow a tree first, then decide.

60
none yetprimes found so far
not finishedprime factorization

60 has not been split yet. Press Next split to divide it by its smallest prime factor.

Show me the steps
  1. 60 = 2 × 30. The 2 is prime, so it stops. 30 is still composite.
  2. 30 = 2 × 15. Another prime 2 stops, 15 continues.
  3. 15 = 3 × 5. Both are prime, so the tree is finished.
  4. The primes are 2, 2, 3, 5: 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.
  5. Try 64: five splits, every one gives a 2, so 64 = 2⁶.
  6. Try a prime such as 97: it cannot be split, so it is its own prime factorization.

Every whole number greater than 1 is a product of primes in exactly one way (apart from the order). 60 = 2 × 2 × 3 × 5, and nothing else.

Index form writes repeats as powers: 2 × 2 × 3 × 5 = 2² × 3 × 5. It is shorter, and it makes factors, multiples and comparisons easier to see.

Different first splits, the same primes

Choose a number, then choose how to split it first. The two sides are broken into primes and put back together.

60
2 × 30
2 × 30 and 3 × 20two first splits
2² × 3 × 5prime factorization of both

60 = 2 × 30 gives the primes 2, 2, 3, 5. A different first split, 60 = 3 × 20, gives 3, 2, 2, 5. Sorted, both are 2, 2, 3, 5, so 2² × 3 × 5.

Show me the steps
  1. 60 = 2 × 30 and 60 = 3 × 20. The first gives 2, then 2, 3, 5. The second gives 3, then 2, 2, 5. Together both are 2, 2, 3, 5.
  2. 60 = 4 × 15 gives 2, 2 then 3, 5. The next split, 60 = 5 × 12, gives 5 then 2, 2, 3. Both are 2, 2, 3, 5 once sorted.
  3. Change to 72 and choose 6 × 12: it gives 2, 3 then 2, 2, 3. The other row, 8 × 9, gives 2, 2, 2 then 3, 3. Both are 2³ × 3².

A common mistake: stopping too early. 60 = 6 × 10 is not a prime factorization, because 6 and 10 are not prime. Keep splitting until every number at the end of a branch has a thick ring.

Worked example: the ladder method

Write 84 as a product of prime factors. Divide by the smallest prime that goes in, again and again.

  1. 84 ÷ 2 = 42.
  2. 42 ÷ 2 = 21.
  3. 21 ÷ 3 = 7, and 7 is prime.
  4. The divisors are 2, 2, 3 and 7: 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7.

Your go. The first two steps are given. Write 180 as a product of prime factors.

  1. 180 ÷ 2 = 90.
  2. 90 ÷ 2 = 45.
  3. Now continue with 45 yourself, then check, then write the answer in index form.
Check the answer

45 ÷ 3 = 15, 15 ÷ 3 = 5, and 5 is prime. The primes are 2, 2, 3, 3 and 5: 180 = 2² × 3² × 5.

Back to the coach

A team size for 60 players uses 0, 1 or 2 of the twos, 0 or 1 of the three and 0 or 1 of the five. That is 3 × 2 × 2 = 12 choices, so there are 12 team sizes: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. The primes show them all at once.

Check before you go on

1. Which of these is 36 written as a product of prime factors?

Your turn

Start at Level 1 and go as far as you can. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. 60 = 2² × 3 × 5. How many prime factors does 60 have, counting repeats?

3. Work out 2³ × 3 × 5.

Level 2 · Building

4. What number has the prime factorization 2² × 3² × 5?

5. 72 = 2³ × 3². How many times does the prime 2 appear in the prime factorization of 72?

6. Which of these is the prime factorization of 84?

Level 3 · Stretch

7. What is the smallest whole number you can multiply 12 by to get a perfect square?

8. A coach has 72 players and wants equal teams. 72 = 2³ × 3². How many different team sizes are possible, counting a team of 1 and a team of all 72?

Something to think about

A number has the prime factorization 2³ × 5² × 7. Without multiplying it out, is it divisible by 14? By 9? By 100? Explain each one.

One way to think about it

14: yes, because 14 = 2 × 7 and both primes are in the factorization. 9: no, because 9 = 3 × 3 and there is no 3. 100: yes, because 100 = 2² × 5² and the factorization has at least two 2s and two 5s.

Exit ticket

9. Priya writes 24 = 2 × 3 × 4 and says this is the prime factorization of 24. Which reply is right?