Skip to content

Skip counting by 2s, 5s and 10s

Number theory · level 1, lesson 4 of 6

A question to start

Count by 5s: 5, 10, 15, 20, 25, ... Keep going as long as you like. Will you ever say the number 42?

What about 100? What about 1,000? What about 1,003?

You could count and count and count to find out. But there is a much faster way. By the end of this lesson you will be able to look at any number and say right away whether counting by 5s will land on it. The same goes for counting by 2s and by 10s.

The idea

What skip counting is

When you count 1, 2, 3, 4, 5, ... you say every number. When you skip count, you jump over some numbers and only say some of them. Counting by 2s means saying every second number: 2, 4, 6, 8, ... Counting by 5s means saying every fifth number: 5, 10, 15, 20, ...

Skip counting is a fast way to count things that come in groups. Socks come in pairs, so count them by 2s. Fingers come in hands of 5, so count them by 5s. Coins worth 10 cents, so count them by 10s.

Counting by 2s on the number line

Start at 0. Hop 2 steps each time. Where do you land?

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

You land on 2, 4, 6, 8, 10, 12, 14, 16, 18, 20. Those are the even numbers, the ones from lesson 1 where everything has a partner.

That makes sense. Each hop of 2 is one more pair. After 3 hops you have 3 pairs, which is 6. After 7 hops you have 7 pairs, which is 14. Counting by 2s is counting pairs.

So here is the answer for 2s: counting by 2s from 0 lands on a number exactly when the number is even. It lands on 36 (it ends in 6). It never lands on 37 (it ends in 7); it jumps right over it, from 36 to 38.

Counting by 5s and by 10s

Now hop by 5s on the first line below, and by 10s on the second line.

0 5 10 15 20 25 30 5s 0 5 10 15 20 25 30 10s

Counting by 5s lands on 5, 10, 15, 20, 25, 30. Counting by 10s lands on 10, 20, 30. Look closely: every number that the 10s land on, the 5s land on too. Why? Because two hops of 5 make one hop of 10. 5 and 5 make 10.

Spotting the landing numbers on a number chart

A number chart puts the numbers in rows of ten. Here are 1 to 30. The numbers you land on when you count by 5s are coloured: blue and orange. The orange ones are also the numbers you land on when you count by 10s.

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

Something wonderful happens. The landing numbers make straight columns!

  • Counting by 10s lands on the last column: 10, 20, 30. These numbers all end in 0.
  • Counting by 5s lands on two columns: the middle one and the last one. These numbers end in 5 or 0.

If you count by 2s on the same chart, you land on the 2 column, the 4 column, the 6 column, the 8 column and the 10 column: five columns, all ending in 0, 2, 4, 6 or 8.

Why the columns are straight

Each row of the chart has ten numbers. So going down one row adds 10.

  • Counting by 10s: one hop of 10 goes straight down one row. So you stay in the same column for ever.
  • Counting by 5s: two hops of 5 make 10, so every second hop lands straight below a number you landed on before. The 5s keep landing in the same two columns, the 5 column and the 10 column.
  • Counting by 2s: five hops of 2 make 10 (2, 4, 6, 8, 10), so the 2s keep landing in the same five columns.

The number 10 is a good friend of 2 and 5. 2 fits into 10 exactly (five times), and 5 fits into 10 exactly (twice). That is why the last digit is enough to tell where counting by 2s, 5s or 10s will land.

The landing rules

Counting by Lands on numbers ending in Examples
2s 0, 2, 4, 6, 8 14, 58, 300
5s 0 or 5 35, 90, 1,005
10s 0 40, 170, 1,000

Now we can answer the question from the start. Counting by 5s never lands on 42 (it ends in 2): it goes 40, 45 and jumps right over 42. It does land on 100 and on 1,000 (they end in 0). It does not land on 1,003 (it ends in 3): it goes from 1,000 to 1,005.

Counting with hands

Hold up one hand: 5 fingers. Two hands: 10. To count the fingers of a group of people, count by 5s, one hand at a time.

5 10 15 20 one hand, two hands, three hands, four hands

Four hands: 5, 10, 15, 20. That is two people's hands, so two people have 20 fingers.

Counting with nickels

A nickel is a coin worth 5 cents, in Canada and in the United States. To find how much a pile of nickels is worth, count by 5s, one nickel at a time.

5 5 5 10 5 15 5 20 5 25 5 30 cents so far

Six nickels are 5, 10, 15, 20, 25, 30: 30 cents. A dime is worth 10 cents, so you count dimes by 10s: three dimes are 10, 20, 30, also 30 cents. So six nickels are worth the same as three dimes. That is the "two 5s make a 10" idea again.

Worked examples

Example 1. Socks by 2s. There are 9 pairs of socks on the line. How many socks is that?

  1. Each pair is 2 socks, so count by 2s, one count for each pair.
  2. 2, 4, 6, 8, 10, 12, 14, 16, 18. That is 9 counts.
  3. So there are 18 socks.

Example 2. Nickels in a jar. Zoe has 7 nickels. How many cents is that?

  1. Count by 5s, one count for each nickel: 5, 10, 15, 20, 25, 30, 35.
  2. That is 7 counts, so Zoe has 35 cents.
  3. Check the last digit: 35 ends in 5, so counting by 5s really can land there. ✓

Example 3. Will it land? Counting by 5s from 0, which of these numbers will you say: 55, 72, 80, 101?

  1. Look at each last digit. You need a 0 or a 5.
  2. 55 ends in 5: yes. 72 ends in 2: no. 80 ends in 0: yes. 101 ends in 1: no.

Example 4. All three at once. Which numbers between 1 and 30 do you say when you count by 2s and by 5s and by 10s?

  1. Counting by 10s says only 10, 20 and 30.
  2. Each of those ends in 0, so counting by 5s says them too, and counting by 2s says them too.
  3. So the answer is 10, 20 and 30: the numbers ending in 0.

Try it

  1. Count by 2s from 0 to 30. How many numbers did you say (not counting 0)?
  2. Count by 10s from 0 to 100. How many numbers did you say?
  3. How many fingers do 3 people have altogether? Count by 5s, one hand at a time.
  4. Sam has 5 nickels. Mo has 2 dimes. Who has more money?
  5. Counting by 2s from 0, do you say 49? Do you say 64?
  6. Counting by 5s from 0, which of these do you say: 25, 52, 60, 99, 115?
  7. Count by 5s, but start at 3: 3, 8, 13, ... What do you notice about the last digits?
  8. Which numbers from 1 to 50 do you say both when you count by 2s and when you count by 5s?
Answers
  1. 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30: 15 numbers. (30 is 15 pairs.)
  2. 10, 20, 30, 40, 50, 60, 70, 80, 90, 100: 10 numbers.
  3. 3 people have 6 hands. Count by 5s six times: 5, 10, 15, 20, 25, 30 fingers.
  4. Sam: 5, 10, 15, 20, 25, so 25 cents. Mo: 10, 20, so 20 cents. Sam has more, by 5 cents.
  5. No for 49: it ends in 9, an odd digit, so the 2s jump from 48 to 50. Yes for 64: it ends in 4.
  6. 25, 60 and 115. They end in 5, 0 and 5. 52 ends in 2 and 99 ends in 9, so the 5s jump over them.
  7. 3, 8, 13, 18, 23, 28, ... The last digits go 3, 8, 3, 8, taking turns. On the number chart these land in two straight columns, the 3 column and the 8 column. Each pair of hops adds 10, so you go straight down.
  8. 10, 20, 30, 40 and 50. Counting by 2s needs a last digit of 0, 2, 4, 6 or 8. Counting by 5s needs a last digit of 0 or 5. The only last digit in both lists is 0.

Common mistake

Skipping a count, or counting a number twice. When skip counting, it is easy to say "15, 20, 30" and miss 25. Touch each thing (each sock, each hand, each nickel) as you say its number, so every count matches one thing.

Thinking "ends in 5" means "counting by 5s only". 25 ends in 5, so counting by 5s lands on it. But 20 ends in 0 and counting by 5s lands on it too. The 5s land on numbers ending in 5 or 0.

Where it leads