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Divisibility by 2, 5 and 10

Number theory · level 3, lesson 1 of 6

A question to start

A friend writes a very long number on a card, 40 digits long. She covers up every digit except the last one, which is a 5. Then she asks you two questions:

  • If you had that many marbles, could you pack them in bags of 5 with none left over?
  • Could you pack them in bags of 2 with none left over?

You cannot see 39 of the 40 digits. It seems impossible to answer. But you can answer both questions straight away, and be completely sure: yes for bags of 5, no for bags of 2. This lesson shows why the hidden digits do not matter at all.

The idea

Some words we will use all the time

When you share a number of things into equal groups, two things can happen.

  • The sharing works out exactly, with nothing left over. Then we say the number is divisible by the group size. 20 is divisible by 5, because 20 marbles make 4 bags of 5 exactly.
  • Some are left over. What is left over is called the remainder. 23 is not divisible by 5: 23 marbles make 4 bags of 5, with 3 left over. The remainder is 3.

If 20 is divisible by 5, we also say:

  • 5 divides 20, or
  • 5 is a factor of 20, or
  • 20 is a multiple of 5 (it is in the 5 times table).

All four sentences mean the same thing. Mathematicians switch between them a lot, so it is worth getting used to all of them.

A divisibility test is a quick way to find out whether a number is divisible by another, without doing the whole division. This level is full of them. But we will not just learn the tests. For each one we will find the reason it works, because a test you understand is a test you cannot forget or mix up.

Numbers are made of tens

Think about how we write numbers. The number 47 means 4 tens and 7 ones. The number 3,658 means 3 thousands, 6 hundreds, 5 tens and 8 ones. But a hundred is just 10 tens, and a thousand is 100 tens. So 3,658 is the same as 365 tens and 8 ones.

That is true for every whole number: it is some number of tens, plus its last digit. The last digit is the number of loose ones that do not fit into a ten.

Every ten splits into 2s and into 5s

Here is the key picture. Draw a ten as a little block of 10 squares, 2 wide and 5 tall.

4 tens 7 ones each ten = 5 rows of 2 = 2 columns of 5

Look at one blue block of ten:

  • It has 5 rows of 2. So a ten always splits into pairs, with none left over.
  • It has 2 columns of 5. So a ten always splits into fives, with none left over.
  • And of course a ten is one group of 10.

So every ten is divisible by 2, by 5 and by 10. That is because 10 = 2 × 5.

Now take any number, say 47. The 4 blue tens split perfectly into pairs. Whether 47 splits into pairs depends only on the 7 loose ones. Seven ones make 3 pairs and 1 left over (the green square). So 47 is odd, and the remainder when you divide 47 by 2 is 1.

It does not matter whether there are 4 tens, or 365 tens, or a million tens. The tens never cause a problem. Only the loose ones, the last digit, can leave something over.

The three tests

Divisible by 2. A number is divisible by 2 when its last digit is divisible by 2: that is, when it ends in 0, 2, 4, 6 or 8. These numbers are called even. Numbers ending in 1, 3, 5, 7 or 9 are odd.

Divisible by 5. A number is divisible by 5 when its last digit is divisible by 5: when it ends in 0 or 5.

Divisible by 10. A number is divisible by 10 when its last digit is 0.

Back to the card with 40 digits ending in 5. The 39 hidden digits only tell you how many tens there are, and tens always split into 5s and into 2s. The last digit, 5, splits into one 5 exactly, but into 2s it leaves 1. So the number is divisible by 5 and not by 2, whatever the hidden digits are.

Finding the remainder too

The same picture tells you the remainder, not just yes or no. The tens leave nothing, so the remainder comes only from the last digit.

  • Dividing by 10: the remainder is just the last digit. 3,658 ÷ 10 leaves 8.
  • Dividing by 5: look at the last digit. If it is 0 to 4, that is the remainder. If it is 5 to 9, take away 5. 3,658 ends in 8, and 8 − 5 = 3, so 3,658 ÷ 5 leaves remainder 3.
  • Dividing by 2: even numbers leave 0 and odd numbers leave 1.

Why this does not work for 3

It is tempting to think "the last digit tells you everything" for every number. Try it with 3. The numbers 12 and 22 both end in 2. But 12 = 4 × 3 is divisible by 3, and 22 is not (22 = 7 × 3 + 1).

What went wrong? A ten does not split into threes: 10 = 3 + 3 + 3 + 1. Every ten leaves one over. So for 3 the tens matter, and the last digit alone cannot tell you the answer. (There is a different test for 3, and it is a beautiful one, in lesson 3.)

The last-digit test works exactly for the numbers that divide 10: 1, 2, 5 and 10. For each of those, a ten splits perfectly.

Worked examples

1. Test 7,385 for 2, 5 and 10.

  • Last digit 5. Is 5 even? No. So 7,385 is not divisible by 2 (it is odd).
  • Last digit 5 is 0 or 5, so 7,385 is divisible by 5. (In fact 7,385 = 5 × 1,477.)
  • Last digit is not 0, so it is not divisible by 10.

2. Find the remainder when 4,629 is divided by 5.

The tens (462 of them) leave nothing. The last digit is 9, which is 5 + 4. So the remainder is 4. Check: 5 × 925 = 4,625, and 4,629 − 4,625 = 4. ✓

3. A missing digit. The number 63_ has its last digit missing.

  • Which digits make it divisible by 2? Any even digit: 630, 632, 634, 636, 638.
  • Which make it divisible by 5? 630 and 635.
  • Which make it divisible by both 2 and 5? Only 630. A number divisible by 2 and by 5 must end in a digit that is even and is 0 or 5. Only 0 is both. That is why being divisible by 2 and 5 is the same as being divisible by 10.

4. Counting multiples. How many numbers from 1 to 100 are divisible by 5?

The multiples of 5 end in 0 or 5. In each block of ten (1 to 10, 11 to 20, ...) there are exactly two: one ending in 5 and one ending in 0. There are 10 blocks, so there are 20. (Check: 100 ÷ 5 = 20.)

5. Odd plus odd. Is 3,457 + 1,289 even or odd? You do not need to add. The last digit of the answer comes from 7 + 9 = 16, so it ends in 6: even. (The full answer is 4,746.)

Try it

  1. Which of these are divisible by 2? By 5? By 10? 340, 785, 1,002, 9,995, 56,000.
  2. Find the remainder when 8,437 is divided by (a) 10, (b) 5, (c) 2.
  3. Fill the gap in 47_ so that the number is divisible by 5 but not by 2.
  4. A number is divisible by 10. Can it be odd? Explain using the picture of tens.
  5. How many numbers from 1 to 200 are divisible by 2 but not by 10?
  6. A farmer has 2,345 eggs and packs them in boxes of 10. How many full boxes? How many eggs are left? If she uses boxes of 5 instead, how many are left?
  7. The number 12 ends in 2 and is divisible by 4. Does every number ending in 2 divide by 4? Find a number that shows the answer.
  8. Explain why a number that ends in 5 can never be divisible by 2, however many digits it has.
Answers
  1. Divisible by 2 (ends in 0, 2, 4, 6, 8): 340, 1,002, 56,000. Divisible by 5 (ends in 0 or 5): 340, 785, 9,995, 56,000. Divisible by 10 (ends in 0): 340 and 56,000.
  2. (a) The last digit: 7. (b) 7 = 5 + 2, so 2. (c) It is odd, so 1. Check (b): 5 × 1,687 = 8,435, and 8,437 − 8,435 = 2.
  3. It must end in 0 or 5 to divide by 5, and must be odd so it does not divide by 2. So the digit is 5, making 475.
  4. No. A number divisible by 10 ends in 0. Its tens split into pairs, and the 0 loose ones leave nothing, so it is even.
  5. From 1 to 200 there are 100 even numbers. Of those, the ones ending in 0 are 10, 20, ..., 200: there are 20. So 100 − 20 = 80 numbers are even but not divisible by 10.
  6. 2,345 is 234 tens and 5 ones, so 234 full boxes of 10 and 5 eggs left. In boxes of 5 the last digit 5 makes one more box exactly, so 0 left (469 boxes).
  7. No. 22 ends in 2, but 22 ÷ 4 = 5 remainder 2. For 4, a ten does not split into fours (10 = 4 + 4 + 2), so the tens matter. The real test for 4 is in the next lesson.
  8. All the tens split into pairs. The last digit is 5, which is 2 pairs and 1 left over. So the whole number has exactly 1 left over when you make pairs. It does not matter how many tens there are.

Common mistake

Using the last digit for the wrong number. The last-digit test works only for 2, 5 and 10 (and 1), because only those numbers divide 10 exactly. "It ends in 3, so it divides by 3" is wrong: 13 ends in 3 and is not divisible by 3, and 21 ends in 1 and is.

Mixing up "divides" and "is divisible by". "5 divides 20" and "20 is divisible by 5" say the same thing, with the numbers in opposite places. The small number divides the big one. "20 divides 5" would be wrong.

Where it leads