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Multiples

Number theory · level 2, lesson 2 of 6

A question to start

Two frogs start on the same lily pad, at 0. The orange frog always jumps 2 pads. The blue frog always jumps 3 pads. They jump along the same row of lily pads, as far as they like.

Which pads do both frogs land on? Is there a pattern? And here is a harder one: if the frogs jumped 4 pads and 6 pads instead, would they meet at 24 first, because 4 times 6 is 24?

Make a guess. By the end of the lesson you will know, and you will know a quick way to find out for any two frogs.

The idea

What a multiple is

When you skip count by 3s from 0, you say 3, 6, 9, 12, 15, ... These are the multiples of 3.

A multiple of a number is what you get when you take that number some whole number of times. 12 is a multiple of 3, because 12 is 3 taken 4 times: 3 + 3 + 3 + 3 = 12, or 4 × 3 = 12. (The sign × means "times", or "groups of".)

The multiples of a number are the numbers in its times table, written as a list.

Number Its multiples (the first ten)
2 2, 4, 6, 8, 10, 12, 14, 16, 18, 20
3 3, 6, 9, 12, 15, 18, 21, 24, 27, 30
4 4, 8, 12, 16, 20, 24, 28, 32, 36, 40
5 5, 10, 15, 20, 25, 30, 35, 40, 45, 50
10 10, 20, 30, 40, 50, 60, 70, 80, 90, 100

A few things to notice:

  • The multiples of 2 are the even numbers. You knew that from level 1: counting by 2s is counting pairs.
  • The list never ends. You can always add one more jump. So every number has infinitely many multiples, which means "more than you could ever count".
  • Every number is a multiple of itself (1 times) and a multiple of 1. 7 is 7 taken once, and 7 is 1 taken 7 times.

A multiple means a leftover of zero

Here is another way to see it. 12 is a multiple of 3 because 12 counters share among 3 friends with nothing left over: each gets 4. And 12 counters make a rectangle of 3 rows of 4. So these three ideas are really one:

  • 12 is a multiple of 3 (it is in the 3 times table),
  • 12 shares fairly among 3 (the leftover is 0),
  • 12 counters make a rectangle with 3 rows.

And 14 is not a multiple of 3: 14 among 3 friends is 4 each with 2 left over. The skip count goes 12, 15, and jumps over 14.

Multiples of 3 on a hundred chart

A hundred chart has the numbers 1 to 100 in rows of ten. Shade the multiples of 3.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100

They make slanting stripes, going down and to the left. Why?

Going down one row adds 10. Going left one square takes away 1. So going down-and-left adds 10 and takes away 1: altogether it adds 9. And 9 is 3 + 3 + 3, three more jumps of 3. So from any multiple of 3, the square down-and-left is also a multiple of 3. The stripe keeps going.

Compare with the multiples of 5 and 10 from Skip counting: they made straight columns, because going down one row adds 10, and 10 is a multiple of 5. Each number paints its own picture on the chart. Multiples of 9 make slanting lines too: 9, 18, 27, 36, ... Each one is one row down and one square left of the one before, because adding 9 is the same as adding 10 and taking away 1.

Common multiples: where the frogs meet

Back to the frogs. The orange frog jumps by 2 (above the line). The blue frog jumps by 3 (below the line).

0 2 3 4 6 8 9 10 12 14 15 16 18 20 21 22 24 2s 3s
  • The orange frog lands on 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, ...
  • The blue frog lands on 3, 6, 9, 12, 15, 18, 21, 24, ...

They both land on 6, 12, 18, 24, ... (the green dots). A number that is a multiple of two numbers at once is a common multiple of them. "Common" means "shared", like a common room that everybody shares.

The common multiples of 2 and 3 are 6, 12, 18, 24, 30, ...: the multiples of 6! The frogs meet every 6 pads.

Why every 6? The first time they meet is at 6. From there, both frogs are back "in step", just like at the start. The orange frog needs 3 more jumps to go 6 more pads, and the blue frog needs 2 more jumps to go 6 more pads. So they meet again at 12, and again at 18, and so on. Once you know the first meeting place, you know them all: just keep adding it.

The smallest common multiple

The first meeting place, the smallest common multiple (not counting 0), is special. For 2 and 3 it is 6. Here is a way to find it for any two numbers: write the multiples of the bigger number, and stop at the first one that is also a multiple of the smaller number.

Now try the frogs that jump 4 and 6.

  • Multiples of 6: 6, 12, ... Is 6 a multiple of 4? No. Is 12 a multiple of 4? Yes! 4, 8, 12.
  • So the frogs first meet at 12, not at 24.

They do meet at 24 too, but that is their second meeting. Multiplying the two numbers always gives a common multiple (4 times 6 is a multiple of 4 and of 6), but not always the smallest. For 2 and 3 the smallest one, 6, happens to be 2 × 3. For 4 and 6 it is smaller, because 4 and 6 already share a 2 inside them: 4 is 2 × 2 and 6 is 2 × 3. You will learn exactly when this happens in level 4.

Worked examples

Example 1. Is it a multiple? Is 35 a multiple of 5? Is 35 a multiple of 4?

  1. Counting by 5s lands on numbers ending in 0 or 5. 35 ends in 5, so yes. (It is 7 × 5.)
  2. Count by 4s: 4, 8, 12, 16, 20, 24, 28, 32, 36. The count goes from 32 to 36 and jumps over 35. So no. (35 among 4 friends is 8 each with 3 left over.)

Example 2. Multiples between two numbers. List the multiples of 7 between 20 and 50.

  1. Count by 7s: 7, 14, 21, 28, 35, 42, 49, 56.
  2. Keep the ones between 20 and 50: 21, 28, 35, 42, 49.

Example 3. Lights blinking together. One light blinks every 4 seconds. Another blinks every 5 seconds. They blink together at the start. When do they next blink together?

  1. The first light blinks at 4, 8, 12, 16, 20, 24, ... seconds.
  2. Write the multiples of the bigger number, 5: 5, 10, 15, 20. Check each one in the 4s list. 20 is there.
  3. So they blink together again after 20 seconds, then 40, then 60, and so on.

Example 4. Buses. Bus A comes every 6 minutes and bus B every 9 minutes. They both leave the station at 8 o'clock. When are they next there together?

  1. Multiples of 9: 9, 18, ... Is 9 a multiple of 6? No. Is 18? 6, 12, 18: yes.
  2. So they are next there together 18 minutes after 8 o'clock. (Not 54 minutes, which is 6 × 9; that is a later meeting.)

Try it

  1. Write the first eight multiples of 6.
  2. Is 27 a multiple of 3? Of 4? Of 9?
  3. On a hundred chart, shade the multiples of 4. What shape do they make?
  4. List the multiples of 8 that are smaller than 50.
  5. Find three common multiples of 2 and 5. What do they have in common?
  6. Find the smallest common multiple of 3 and 4.
  7. Find the smallest common multiple of 6 and 8. Is it 48?
  8. Three frogs jump 2, 3 and 4 pads, all starting at 0. Where do all three first land together?
Answers
  1. 6, 12, 18, 24, 30, 36, 42, 48.
  2. Multiple of 3: yes (9 × 3 = 27). Of 4: no (the 4s go 24, 28 and jump over 27; it is also odd, and all multiples of 4 are even). Of 9: yes (3 × 9 = 27).
  3. 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ... They make a zig-zag pattern of columns. In the 1st, 3rd, 5th, ... rows they sit in the columns ending in 4 and 8 (4, 8; 24, 28; ...). In the 2nd, 4th, 6th, ... rows they sit in the columns ending in 2, 6 and 0 (12, 16, 20; 32, 36, 40; ...). The pattern repeats every two rows, because going down two rows adds 20, and 20 is 5 × 4.
  4. 8, 16, 24, 32, 40, 48.
  5. For example 10, 20, 30. They all end in 0: they are the multiples of 10. (Counting by 2s needs a last digit 0, 2, 4, 6, 8; counting by 5s needs 0 or 5; both means 0.)
  6. Multiples of 4: 4, 8, 12. Is 4 a multiple of 3? No. Is 8? No. Is 12? Yes. So 12.
  7. Multiples of 8: 8, 16, 24. Is 8 a multiple of 6? No. Is 16? No (6, 12, 18). Is 24? Yes (4 × 6). So the smallest is 24, not 48. 48 is a common multiple, but it is the second one.
  8. The frogs jumping 2 and 4: every multiple of 4 is also a multiple of 2, so they meet at 4, 8, 12, ... Now add the 3s frog: the first multiple of 4 that is also a multiple of 3 is 12. So all three first meet at 12.

Common mistake

Always multiplying to find where they meet. 4 × 6 = 24, but 4 and 6 first meet at 12. Multiplying gives a common multiple, but check smaller ones by listing the multiples of the bigger number.

Mixing up "multiple" and "factor". The multiples of 3 are bigger (or the same): 3, 6, 9, 12, ... A factor is a number that goes into another one, so it is smaller (or the same). You will meet factors in the next lesson. A trick: multiples are more.

Where it leads