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Factors with arrays

Number theory · level 2, lesson 3 of 6

A question to start

A baker has 24 cupcakes. She wants to put them on a tray in neat rows, every row the same length. How many different trays could she use?

She could use 2 rows of 12. Or 3 rows of 8. Are there more? How can she be sure she has found them all, and not missed one? Many people find a few and then stop, not knowing if more are hiding. In this lesson you will learn a way to find every answer and to know you are finished.

The idea

Arrays

An array is a rectangle of things set out in rows and columns, like eggs in an egg box, seats in a cinema, or cupcakes on a tray. Rows go across; columns go up and down. Every row has the same number of things.

We describe an array by its number of rows and the number in each row. An array with 3 rows of 4 is a "3 by 4" array. It holds 3 × 4 = 12 things.

In level 1 you built Rectangle numbers. Arrays are the same idea, but now we also allow the long, thin array with just 1 row. You will see why that helps.

All the arrays for 12

Here is every array you can make with 12 counters.

1 by 12 2 by 6 3 by 4
  • 1 row of 12: 1 × 12 = 12.
  • 2 rows of 6: 2 × 6 = 12.
  • 3 rows of 4: 3 × 4 = 12.

(4 rows of 3, 6 rows of 2 and 12 rows of 1 are the same arrays turned on their side.)

Factors

The numbers that make the sides of the arrays are called the factors of 12. A factor of a number is a number that goes into it exactly, with nothing left over.

From the arrays, the factors of 12 are 1, 2, 3, 4, 6 and 12.

Here are three ways to say the same thing:

  • 3 is a factor of 12 (it is the side of an array of 12).
  • 12 is a multiple of 3 (it is in the 3 times table, from the last lesson).
  • 12 shared among 3 friends leaves 0 left over (from Leftovers).

Factors and multiples are two ends of the same rope. Factors go in. Multiples come out.

5 is not a factor of 12: 12 among 5 friends is 2 each with 2 left over, and 5 rows can never make an array of 12.

Factor pairs

Each array gives two factors at once: the number of rows and the number in each row. These two factors make a factor pair: two numbers that multiply to make the number.

The factor pairs of 12 are:

  • 1 and 12 (1 × 12 = 12)
  • 2 and 6 (2 × 6 = 12)
  • 3 and 4 (3 × 4 = 12)

So factors come in pairs. When you find one factor, you get its partner free: just ask "how many in each row?" If you know 2 is a factor of 12, then 12 among 2 rows gives 6 in each row, so 6 is a factor too.

The factor rainbow

Write the factors of 12 in order and join each factor to its partner with an arc. You get a factor rainbow.

1 2 3 4 6 12

The rainbow shows something beautiful. The small factors are partners with the big factors. The smallest, 1, is partners with the biggest, 12. The next smallest, 2, is partners with the next biggest, 6. And so on, until the partners meet in the middle.

Why? If you make the rows fewer, each row has to be longer to hold the same number of counters. As one factor goes up, its partner comes down. So the pairs "squeeze in" towards the middle.

Finding all the factors, and knowing you are done

This gives a plan that never misses a factor.

  1. Try 1, then 2, then 3, and so on, as the number of rows.
  2. Each time it works (nothing left over), write down the factor and its partner.
  3. Stop when the number of rows gets as big as the number in each row. After that, every array is one you have already found, turned on its side.

Let us find all the trays for 24 cupcakes.

Rows Works? In each row Factor pair
1 yes 24 1 and 24
2 yes (24 is even) 12 2 and 12
3 yes (3 × 8 = 24) 8 3 and 8
4 yes (4 × 6 = 24) 6 4 and 6
5 no (5 × 4 = 20, 4 left over)
6 yes, but 6 rows of 4 is 4 rows of 6 turned. Stop.

So the factors of 24 are 1, 2, 3, 4, 6, 8, 12 and 24: eight factors, in four pairs. The baker has four trays: 1 by 24 (a very long tray!), 2 by 12, 3 by 8 and 4 by 6.

How do we know we are done? Every array has a short side and a long side (or two equal sides). Turn it so that the short side is the number of rows. We tried every short side from 1 up to 5. A short side of 5 or more would need at least 5 rows of 5, which is 25 cupcakes, and we only have 24. So there is nothing left to try.

A pair that is lonely

Find the factors of 36 the same way: 1 and 36, 2 and 18, 3 and 12, 4 and 9, then 5 does not work, then 6 and 6!

1 2 3 4 6 9 12 18 36

The 6 pairs with itself, because 6 rows of 6 is a square. In the rainbow it stands alone in the middle, with no arc. So 36 has 9 factors: 1, 2, 3, 4, 6, 9, 12, 18, 36. That is an odd number of factors. You will find out exactly which numbers do this in Square numbers and the locker puzzle.

Worked examples

Example 1. Factors of 18.

  1. 1 row of 18: pair 1 and 18.
  2. 2 rows: 18 is even, so 9 in each row. Pair 2 and 9.
  3. 3 rows: 3 × 6 = 18. Pair 3 and 6.
  4. 4 rows: 4 × 4 = 16, 2 left over. No.
  5. 5 rows: 5 × 3 = 15, 3 left over. No. And 5 rows would hold fewer than 5 in each row, so stop.
  6. The factors of 18 are 1, 2, 3, 6, 9, 18.

Example 2. Factors of 20.

  1. 1 × 20, 2 × 10, 4 × 5. (3 does not work: 3 × 6 = 18, 2 left over.)
  2. 5 rows would give 4 in each row: that is 4 × 5 turned. Stop.
  3. The factors of 20 are 1, 2, 4, 5, 10, 20.

Example 3. Factors of 7.

  1. 1 × 7. That is a pair.
  2. 2 rows: 7 is odd, 1 left over. No. 3 rows: 3 × 2 = 6, 1 left over. No.
  3. 3 rows would already hold fewer than 3 in each row (only 2), so stop.
  4. The only factors of 7 are 1 and 7: just one array, the line. 7 is a "line-only" number from level 1.

Example 4. Is 6 a factor of 42?

  1. Count by 6s: 6, 12, 18, 24, 30, 36, 42. Yes, 42 is there: 7 × 6 = 42.
  2. So yes, and its partner is 7.

Try it

  1. Draw all the arrays for 8. What are the factors of 8?
  2. List the factors of 10.
  3. List the factors of 16. Draw its factor rainbow. What do you notice in the middle?
  4. Is 4 a factor of 30? Is 5? Is 6?
  5. 3 is a factor of 27. What is its partner?
  6. List all the factors of 30, in pairs.
  7. A teacher has 28 chairs to set out in equal rows. What are all the ways? (Count 4 rows of 7 and 7 rows of 4 as the same.)
  8. Which number below 20 has the most factors?
Answers
  1. 1 by 8 and 2 by 4. The factors are 1, 2, 4, 8.
  2. 1 × 10, 2 × 5. (3 and 4 do not work.) Factors: 1, 2, 5, 10.
  3. 1 × 16, 2 × 8, 4 × 4. (3 does not work: 3 × 5 = 15, 1 left over.) Factors: 1, 2, 4, 8, 16. In the rainbow, 4 stands alone in the middle, because it pairs with itself (4 × 4 = 16). So 16 has an odd number of factors, 5.
  4. 4: no (4 × 7 = 28, 2 left over). 5: yes (5 × 6 = 30). 6: yes (6 × 5 = 30).
  5. 9, because 3 × 9 = 27.
  6. 1 and 30, 2 and 15, 3 and 10, 5 and 6. (4 does not work: 4 × 7 = 28, 2 left over.) Then 6 rows would be 5 × 6 turned, so stop. Factors: 1, 2, 3, 5, 6, 10, 15, 30.
  7. 1 row of 28, 2 rows of 14, 4 rows of 7. (3 does not work: 3 × 9 = 27, 1 left over. 5 does not work: 5 × 5 = 25, 3 left over.) 6 rows: 6 × 4 = 24 and 6 × 5 = 30, so no; and by now the rows are shorter than the number of rows, so stop. Three ways: 1 by 28, 2 by 14, 4 by 7.
  8. 12 and 18 both have 6 factors (1, 2, 3, 4, 6, 12 and 1, 2, 3, 6, 9, 18). No number below 20 has more. (16 has 5 factors; 6, 8, 10, 14 and 15 have 4.)

Common mistake

Forgetting 1 and the number itself. Every number has the long, thin array: 1 row. So 1 and the number itself are always factors. The factors of 12 include 1 and 12.

Forgetting the partner. If you write "the factors of 24 are 1, 2, 3, 4" and stop, you have missed half of them. Each factor you find brings its partner: 24, 12, 8 and 6.

Counting the middle partner twice. In 36, 6 pairs with itself. Write 6 once.

Where it leads