Chapter 5 of 8

All 8 chapters
  1. 1Below zero, readunderstood
  2. 2Calculating with negative numbers, readunderstood
  3. 3Multiples and factors, readunderstood
  4. 4Quick tests for dividing, readunderstood
  5. 5Prime numbers, readunderstood
  6. 6Prime factors, common factors and multiples, readunderstood
  7. 7Squares, cubes and roots, readunderstood
  8. 8Powers of ten, readunderstood

M3.1.5 Chapter 5

Prime numbers

Why can 7 stones go in only one row, but 6 stones can make two rows?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Say if a number up to 100 is prime
  • Find all primes up to 100 by crossing out multiples

Mei has 7 stones. She tries to lay them out in equal rows. She can make 1 row of 7. She cannot make 2 rows, or 3, or 4. Her friend has 6 stones and can make 2 rows of 3.

A prime number has exactly two factors: 1 and itself. 7 is prime, because its only factors are 1 and 7. A composite number has more than two factors. 6 is composite. Its factors are 1, 2, 3 and 6.

Two rectangles of tiles. The first is labelled 11 stones: 1 × 11 only, and it is one row of 11 tiles. The second is labelled 12 stones: 3 × 4, and it is 3 rows of 4 tiles.11 stones: 1 × 11 only12 stones: 3 × 4
11 is prime: it cannot be laid out in more than one row of equal length. 12 is composite.

1 is neither prime nor composite. It has only one factor, which is itself. 2 is prime. It is the only even prime. Every other even number divides by 2, so it has at least three factors.

The primes below 20 are 2, 3, 5, 7, 11, 13, 17 and 19. The other numbers from 4 to 18 are composite.

The numbers 1 to 20 in two rows of ten. 1 and every composite number is shaded out. The primes 2, 3, 5, 7, 11, 13, 17 and 19 are ringed.1234567891011121314151617181920
The primes below 20 are ringed. Every other number is crossed out.

To find every prime up to 100, cross out numbers in a hundred square. This method is called a sieve. Like a kitchen sieve, it keeps the numbers you want and lets the others fall out.

In the pictures, a crossed out number has a line through its cell and is shaded. A prime has a ring round it and bold digits.

Draw it first

Find the primes up to 100 with a sieve

  1. A hundred square holding the numbers 1 to 100 in ten rows of ten. Step 1: 1 is shaded out and 2 is ringed. Step 2: every multiple of 2 after 2 and every multiple of 3 after 3 is shaded out, and 2 and 3 are ringed. Step 3: the multiples of 5 and 7 are shaded out too, and every number left is ringed. These are the primes.123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100

    1 Cross out 1, which is not prime. Ring 2, the first prime.

  2. A hundred square holding the numbers 1 to 100 in ten rows of ten. Step 1: 1 is shaded out and 2 is ringed. Step 2: every multiple of 2 after 2 and every multiple of 3 after 3 is shaded out, and 2 and 3 are ringed. Step 3: the multiples of 5 and 7 are shaded out too, and every number left is ringed. These are the primes.123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100

    2 Cross out every multiple of 2 after 2. Ring 3. Cross out every multiple of 3 after 3.

  3. A hundred square holding the numbers 1 to 100 in ten rows of ten. Step 1: 1 is shaded out and 2 is ringed. Step 2: every multiple of 2 after 2 and every multiple of 3 after 3 is shaded out, and 2 and 3 are ringed. Step 3: the multiples of 5 and 7 are shaded out too, and every number left is ringed. These are the primes.123456789101112131415161718192021222324252627282930313233343536373839404142434445464748495051525354555657585960616263646566676869707172737475767778798081828384858687888990919293949596979899100

    3 Do the same for 5 and 7. Every number left is a prime. Ring them all.

Why stop at 7? The next prime is 11, and 11 × 11 = 121 is more than 100. A composite number up to 100 has a factor of 10 or less, other than 1. That factor is 2, 3, 5 or 7, or divides by one of them. So the sieve has crossed it out.

You can also test one number from 2 to 100 without a sieve. Divide it by 2, 3, 5 and 7. If none of them divides it exactly, it is prime. This test only works up to 100.

A table with the head divide by and remainder. 2 leaves 1, 3 leaves 1, 5 leaves 1, 7 leaves 0. The cell for 7 is shaded, because 7 divides 91 exactly.divide byremainder21315170
The remainders when 91 is divided by 2, 3, 5 and 7.

Worked example

Is 91 prime?

  1. Test 2, 3, 5 and 7. 91 is odd, so 2 fails. For 3, the digit sum is 9 + 1 = 10.
  2. 10 is not in the 3 times table, so 3 fails. 91 does not end in 0 or 5, so 5 fails.
  3. Now try 7. 91 ÷ 7 = 13, with nothing left over. So 7 is a factor and 91 is not prime.

Answer: No, 91 = 7 × 13.

Worked example

Is 59 prime?

  1. 59 is odd, so 2 fails. For 3, the digit sum is 5 + 9 = 14.
  2. 14 is not in the 3 times table, so 3 fails. 59 ends in 9, so 5 fails.
  3. For 7, look at 7 × 8 = 56 and 7 × 9 = 63.
  4. Neither is 59, so 7 fails. No test finds a factor, so 59 is prime.

Answer: Yes, 59 is prime.

See it three ways

With things

Take 7 stones. Try to make equal rows. Only one row of 7 works. Now take 6 stones. You can make 2 rows of 3, or 3 rows of 2.

As a picture

Two rectangles of tiles. The first is labelled 7 stones: 1 × 7 only, and it is one row of 7 tiles. The second is labelled 6 stones: 2 × 3, and it is 2 rows of 3 tiles.7 stones: 1 × 7 only6 stones: 2 × 3

In symbols

7 has two factors, 1 and 7, so it is prime. 6 has four factors, 1, 2, 3 and 6, so it is composite.

Try it

Make a sieve to 50

You need: a sheet of paper, a pencil, a ruler or a straight edge.

  1. Write the numbers 1 to 50 in rows of 10.
  2. Cross out 1. Ring 2. Cross out every second number after 2.
  3. Ring 3. Cross out every third number after 3. Do the same for 5 and 7.
  4. Ring every number that is left. Count the rings.

What you will see: There are 15 rings. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. After them come 31, 37, 41, 43 and 47.

If you cannot try it, this is what happens.
A square of the numbers 1 to 50 in five rows of ten. 1 and every composite number is shaded out. Every prime is ringed.1234567891011121314151617181920212223242526272829303132333435363738394041424344454647484950

A wrong idea many people have

Every odd number is prime.

Why it feels right: All the even numbers after 2 are composite. So the odd numbers look like the ones that are left.

The test that breaks it: Try 9. 9 = 3 × 3, so it has three factors. Try 15. 15 = 3 × 5. Try 21 and 25 too.

What is true: Odd numbers can be composite. 9, 15, 21, 25 and 27 are all odd and composite. Only the numbers with exactly two factors are prime.

Key words

prime number
a number with exactly two factors, 1 and itself.
composite number
a number with more than two factors.
sieve
a way to find primes by crossing out multiples.

Back to your guess

The answer is B: A prime number has exactly two factors: 1 and itself.

A prime number has exactly two factors, 1 and itself. 9 is odd but 3 divides it, so 9 is not prime. And 2 is prime, even though it is in the 2 times table.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Is 51 prime? Say why.

    Tap your answer.

    1. Not this one. Many odd numbers are not prime. The digit sum of 51 is 6. That is in the 3 times table, so 3 divides 51. Go to M3.1.5. Try another.

    2. Not this one. You left out 3. Test 2, 3, 5 and 7. The digit sum of 51 is 6, so 3 divides 51. Go to M3.1.5. Try another.

    3. Yes, that is right. The digit sum is 5 + 1 = 6, which is in the 3 times table. So 3 divides 51, and 51 = 3 × 17.

    See the answer

    No, 51 = 3 × 17.

    The digit sum is 5 + 1 = 6, which is in the 3 times table. So 3 divides 51, and 51 = 3 × 17.

    • If you chose A: Many odd numbers are not prime. The digit sum of 51 is 6. That is in the 3 times table, so 3 divides 51. Go to M3.1.5.
    • If you chose B: You left out 3. Test 2, 3, 5 and 7. The digit sum of 51 is 6, so 3 divides 51. Go to M3.1.5.

  2. Which of these numbers is prime?

    Tap your answer.

    1. Yes, that is right. Test 2, 3, 5 and 7 on 59. None divides it exactly. The other three each have a factor from that list.

    2. Not this one. 65 ends in 5, so 5 divides it. 65 = 5 × 13. Go to M3.1.5. Try another.

    3. Not this one. You may think every odd number is prime. The digit sum of 57 is 12, so 3 divides it. 57 = 3 × 19. Go to M3.1.5. Try another.

    4. Not this one. 63 is in the 7 times table and in the 9 times table. It has more than two factors. Go to M3.1.5. Try another.

    See the answer

    59

    Test 2, 3, 5 and 7 on 59. None divides it exactly. The other three each have a factor from that list.

    • If you chose B: 65 ends in 5, so 5 divides it. 65 = 5 × 13. Go to M3.1.5.
    • If you chose C: You may think every odd number is prime. The digit sum of 57 is 12, so 3 divides it. 57 = 3 × 19. Go to M3.1.5.
    • If you chose D: 63 is in the 7 times table and in the 9 times table. It has more than two factors. Go to M3.1.5.

  3. Which of these statements is true?

    Tap your answer.

    1. Yes, that is right. Every even number after 2 divides by 2, so it has at least three factors. Only 2 has just two factors, 1 and 2.

    2. Not this one. The number 2 is even, and its only factors are 1 and 2. So 2 is an even prime. Go to M3.1.5. Try another.

    3. Not this one. The number 1 has only one factor, itself. A prime has exactly two factors, so 1 is not prime. Go to M3.1.5. Try another.

    See the answer

    The number 2 is the only even prime.

    Every even number after 2 divides by 2, so it has at least three factors. Only 2 has just two factors, 1 and 2.

    • If you chose B: The number 2 is even, and its only factors are 1 and 2. So 2 is an even prime. Go to M3.1.5.
    • If you chose C: The number 1 has only one factor, itself. A prime has exactly two factors, so 1 is not prime. Go to M3.1.5.

  4. The picture shows a sieve to 30 after 2 and 3. Which numbers do you cross out next?

    The numbers 1 to 30 in three rows of ten. 1 and the multiples of 2 and 3 are shaded out. The numbers 2 and 3 are ringed.123456789101112131415161718192021222324252627282930
    Tap your answer.

    1. Not this one. Every multiple of 4 is also a multiple of 2, so it is already crossed out. Go to M3.1.5. Try another.

    2. Not this one. Every multiple of 6 is also a multiple of 2, so it is already crossed out. Go to M3.1.5. Try another.

    3. Yes, that is right. The first number not crossed out after 3 is 5. It is prime. Ring 5, then cross out the other multiples of 5.

    See the answer

    The multiples of 5

    The first number not crossed out after 3 is 5. It is prime. Ring 5, then cross out the other multiples of 5.

    • If you chose A: Every multiple of 4 is also a multiple of 2, so it is already crossed out. Go to M3.1.5.
    • If you chose B: Every multiple of 6 is also a multiple of 2, so it is already crossed out. Go to M3.1.5.

  5. Why do we cross out only multiples of 2, 3, 5 and 7 up to 100?

    Tap your answer.

    1. Not this one. That sounds right, but it is not the reason. The sieve stops at 7 because 11 × 11 = 121 is more than 100. Go to M3.1.5. Try another.

    2. Yes, that is right. A composite number up to 100 has a factor of 10 or less, other than 1. Every such factor is 2, 3, 5 or 7, or divides by one of them.

    3. Not this one. There are many primes after 7, such as 11, 13 and 17. They just have no new multiples to cross out below 100. Go to M3.1.5. Try another.

    See the answer

    The next prime is 11, and 11 × 11 = 121 is more than 100.

    A composite number up to 100 has a factor of 10 or less, other than 1. Every such factor is 2, 3, 5 or 7, or divides by one of them.

    • If you chose A: That sounds right, but it is not the reason. The sieve stops at 7 because 11 × 11 = 121 is more than 100. Go to M3.1.5.
    • If you chose C: There are many primes after 7, such as 11, 13 and 17. They just have no new multiples to cross out below 100. Go to M3.1.5.

  6. What is the smallest prime factor of 91?

    See the answer

    7

    2, 3 and 5 do not divide 91, but 7 × 13 = 91.

  7. How many primes are there from 1 to 30?

    See the answer

    10 primes

    The primes are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29. That makes 10.

  8. Look at the last picture of the sieve in "Draw it first". Count the ringed numbers. How many primes are there up to 100?

    See the answer

    25 primes

    Count the rings row by row: 4, 4, 2, 2, 3, 2, 2, 3, 2 and 1. They add up to 25.

  9. From an earlier chapter

    Is 8 124 divisible by 3? Use the digit sum.

    See the answer

    Yes, because the digit sum is 15.

    8 + 1 + 2 + 4 = 15, and 15 = 3 × 5. So 8 124 is divisible by 3.

Keep it

Copy this into your exercise book.

A prime has exactly two factors, 1 and itself. 1 is not prime. 2 is the only even prime. There are 25 primes up to 100. For a number from 2 to 100, test with 2, 3, 5 and 7.

Draw: A square of the numbers 1 to 30 in rows of 10. Cross out the composite numbers and ring the primes.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (805 KB). It opens on any phone.