Factors, multiples and primes: Year 7 number systems and place value

Warm-up

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

What is 6 × 7? What is 42 ÷ 6?
Answer
42, then 7. Division undoes multiplication.
What is 12 × 12?
Answer
144. Twelve rows of twelve.
Write 5 × 5 × 5 as a power.
Answer
5³, which equals 125.

Words used in this lesson

factor: a whole number that divides another exactly. For example: the factors of 6 are 1, 2, 3, 6.
multiple: what you get when you multiply by a whole number. For example: the multiples of 6 are 6, 12, 18.
prime number: a number with exactly two factors, 1 and itself. For example: 2, 3, 5, 7, 11.
composite number: a number with more than two factors. For example: 12 is composite.
divisible: can be divided exactly, with no remainder. For example: 12 is divisible by 4.
digit sum: the sum of the digits of a number. For example: the digit sum of 138 is 12.

Rectangles from squares

Twelve seedlings can be planted in a single row of 12, in 2 rows of 6, or in 3 rows of 4. Thirteen seedlings only make one row of 13. A factor of a number is a whole number that divides it exactly, and every rectangle you can make gives a pair of factors.

Predict first: how many rectangles can 12 squares make? And 13? Try the slider on numbers you expect to be awkward.

The number 12

6number of factors

compositethe number is

12 has 6 factors: 1, 2, 3, 4, 6, 12. They make 3 rectangles: 1 × 12, 2 × 6 and 3 × 4.

The figure needs JavaScript. In words: 12 squares can be arranged as 1 by 12, 2 by 6 or 3 by 4, so its factors are 1, 2, 3, 4, 6 and 12. A prime such as 13 only makes the 1 by 13 rectangle, so it has exactly two factors: 1 and itself.

Show me the steps

12: three rectangles, 1 × 12, 2 × 6 and 3 × 4. Six factors, composite.
13: only 1 × 13. Two factors, so 13 is prime.
36: five rectangles, and one is a square, 6 × 6. Nine factors; 6 appears once because it pairs with itself.
59: only 1 × 59, so 59 is prime.
60: six rectangles, 12 factors: the most any number up to 60 has.

A prime number has exactly two factors: 1 and itself. A composite number has more than two. 1 is neither: it has only one factor.

A multiple of a number is what you get by multiplying it by a whole number: the multiples of 6 are 6, 12, 18, 24 and so on. If a divides b, then a is a factor of b and b is a multiple of a.

Quick tests for divisibility

Checking every factor is slow for big numbers. Short tests do the same job: each looks at one part of the number. Here are the tests, and the figure applies each one to any number you choose.

2: the last digit is even (0, 2, 4, 6 or 8).
3: the digit sum is a multiple of 3.
4: the last two digits make a multiple of 4.
5: the last digit is 0 or 5.
6: the number is divisible by both 2 and 3.
8: the last three digits make a multiple of 8.
9: the digit sum is a multiple of 9.
10: the last digit is 0.

The number 138

12digit sum

2, 3, 6divisible by

138 is divisible by 2, 3 and 6: it is even, the digit sum 12 is a multiple of 3, and a number divisible by both 2 and 3 is divisible by 6.

The figure needs JavaScript. In words: a number is divisible by 2 if its last digit is even, by 5 if it ends in 0 or 5, by 10 if it ends in 0, by 3 if its digit sum is a multiple of 3, by 9 if its digit sum is a multiple of 9, by 4 if its last two digits are, by 8 if its last three digits are, and by 6 if it is divisible by both 2 and 3. For 138 the digit sum is 12, so it is divisible by 2, 3 and 6.

Show me the steps

138: even, digit sum 12. Divisible by 2, 3 and 6.
135: ends in 5, digit sum 9. Divisible by 3, 5 and 9; odd, so not by 2 or 6.
120: last digit 0, digit sum 3, last two digits 20, last three digits 120. Divisible by 2, 3, 4, 5, 6, 8 and 10, but not 9.
77: odd, digit sum 14. Not divisible by 2, 3, 4, 5, 6, 8, 9 or 10, though it is divisible by 7.

Why do they work?

For 2, 5 and 10: 10 is a multiple of 2, 5 and 10, so only the last digit can spoil it.
For 4 and 8: 100 is a multiple of 4 and 1000 is a multiple of 8, so only the last two or three digits matter.
For 3 and 9: 10 = 9 + 1 and 100 = 99 + 1, so each place leaves remainder 1 when divided by 9 (or 3). A digit d in that place leaves remainder d, and the remainders add up to the digit sum.

A common mistake: “primes are odd numbers.” 2 is prime and even, and 9, 15 and 21 are odd but have more than two factors. Check the factors, not just the last digit.

Worked example: is 91 prime?

Decide whether 91 is prime.

91 is odd, so 2 is not a factor. The digit sum is 10, not a multiple of 3. It does not end in 0 or 5.
Try 7: 7 × 13 = 91.
91 has a factor other than 1 and itself, so 91 is composite, not prime.

Your go. The first two steps are given. Is 89 prime? If 89 had a factor pair, the smaller factor would be 9 or less, because 10 × 10 is already 100. So test only the primes up to 9.

89 is odd, so 2 is not a factor.
The digit sum is 17, not a multiple of 3, and 89 does not end in 0 or 5.
Now test 7 yourself, then decide, then check.

Check the answer

7 × 12 = 84 and 7 × 13 = 91, so 7 is not a factor. No prime up to 9 divides 89, so 89 is prime.

Check before you go on

1. Which of these numbers is prime?

21 27 29 39

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. How many factors does 18 have?

3. What is the smallest prime number greater than 20?

Level 2 · Building

4. What is the smallest multiple of 6 that is greater than 100?

5. Which of these numbers is divisible by 6?

124 135 138 141

6. I am a prime number between 30 and 50. My digits add up to 10. What number am I?

Level 3 · Stretch

7. The three-digit number 7■4 is divisible by 9. What digit is hidden by the square?

8. What is the smallest whole number that has exactly 6 factors?

Something to think about

Multiply the first six primes: 2 × 3 × 5 × 7 × 11 × 13 = 30030. Add 1 to get 30031. Can 30031 be divided by any of those six primes? What does that suggest about the list of primes?

One way to think about it

30031 leaves remainder 1 when divided by each of 2, 3, 5, 7, 11 and 13, so none of them divides it. Its prime factors must be new primes (30031 = 59 × 509). The same trick works for any list of primes, so the list never ends: there is no largest prime.

Exit ticket

9. Dilnoza says 1 is a prime number because it can only be divided by 1 and itself. Which reply is right?

She is right: 1 is the smallest prime. She is wrong: a prime has exactly two factors, and 1 has only one. She is wrong: 1 is even, so it is not prime. She is right: every number has a factor of 1.