Exercises on multiples and divisors

Although simple, the concepts of multiples and divisors are widely used in mathematics.

The multiples of a number are those that we get by multiplying that number by 0, 1, 2, 3, 4, 5, …, and so on.

The divisors of a number are all those for which the division of the number by them is an exact division, that is, with remainder equal to zero.

Want to learn more about these numbers? check out a list of exercises on multiples and divisors, all of them resolved, step by step, so you can clear all your doubts.

List of exercises on multiples and divisors

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Question 1. Check if 84 is a multiple of:

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Question 2. What are the multiples of 3 between 16 and 35?

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Question 3. What are the multiples of 5 between 123 and 150?

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Question 4. A kit of socks comes with three pairs. If Roberto bought a certain amount of kits, is it possible that he bought 23 pairs of socks?

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Question 5. In the previous question, what are the seven smallest quantities of pairs of socks that Roberto could have bought?

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Question 6. Which numbers below are divisors of 54?

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Question 7. Which of the divisors of 15 are divisors of 25 as well?

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Question 8. What is the number of divisors of:

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Question 9. In how many different ways can we distribute 100 candies into packets that have the same number?

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Question 10. A teacher wants to arrange her 27 students in rows with the same number of students each. In how many ways can she do this?

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Resolution of question 1

Being a multiple of a number is the same as being divisible by that number.

So we have to check, in each case, whether 84 is divisible by the number in question.

a) Yes, because 84 is divisible by 3.
b) Yes, because 84 is divisible by 6.
c) No, because 84 is not divisible by 16.
d) Yes, because 84 is divisible by 21.

Resolution of question 2

We want to find the multiples of 3 between 16 and 35. Among these numbers, the smallest multiple of 3 is 18, as 18 is divisible by 3.

The next multiples can be obtained by adding 3 units to the previous one, so the multiples of 3 between 16 and 35 are: 18, 21, 24, 27, 30 and 33.

Resolution of question 3

Between the numbers 123 and 150, the smallest multiple of 5 is 125, as 125 is divisible by 5.

The next multiples can be obtained by adding 5 units to the previous one. So the multiples of 5 between 123 and 150 are: 125, 130, 135, 140, 145, 150.

Resolution of question 4

It is not possible as the kits come with three pairs of socks and 23 is not a multiple of 3.

Resolution of question 5

They are the multiples of 3, starting with 3 itself, that is: 3, 6, 9, 12, 15, 18, 21, 24

Resolution of question 6

A number a is divisible by a number b only when b is divisible by a.

Thus, we have to check, in each case, whether 54 is divisible by the number in question.

a) Yes, because 54 is divisible by 2.
b) No, because 54 is not divisible by 4.
c) Yes, because 54 is divisible by 9.
d) No, because 54 is not divisible by 11.

Resolution of question 7

First, let's find the divisors of each of the numbers.

D(15) = {1, 3, 5, 15}

D(25) = {1, 5, 25}

So the divisors of 15 that are also divisors of 25 are 1 and 5.

Resolution of question 8

a) To find the number of divisors of a number, we must first do the decomposition into prime factors.

24 | 2
12 | 2
6 | 2
3 | 3
1

Therefore, 24 = 2. 2. 2. 3 = 2³. 3¹

Now, from the exponents of the factors, we determine the number of divisors:

n = (3 + 1). (1 + 1) = 4. 2 = 6

So 24 has 6 divisors.

b) 70 = 2. 5. 7 = 2¹. 5¹. 7¹

n = (1 + 1). (1 + 1). (1 + 1) = 8

c) 582 = 2. 3. 97 = 2¹. 3¹. 97¹

n = (1 + 1). (1 + 1). (1 + 1) = 8

d) 7020 = 2². 3³. 5. 13 = 2². 3³. 5¹. 13¹

n = (2 + 1). (3 + 1). (1 + 1). (1 + 1) = 48

Resolution of question 9

The number of ways we can divide 50 candies into equal amounts is the same number of divisors of 50.

100 = 2. 5²

n = (1 + 1). (2 + 1) = 6

So there are 6 different ways.

The divisors of 50 are: 1, 2, 5, 10, 25 and 50. So the different ways are:

1 pack of 50 candies;
2 packages with 25 candies each;
5 packages with 10 candies each;
10 packages with 5 candies each;
25 packages with 2 candies each;
50 packages with 1 bullet each.

Resolution of question 10

The number of ways we can divide 27 students into rows of the same number is the same number of divisors of 27.

27 = 3³

n = (3 + 1) = 4

So there are 4 different ways.

The divisors of 27 are: 1, 3, 9 and 27. So the different ways are:

1 row with 27 students
3 lines with 9 students each;
9 lines with 3 students each;
27 rows with 1 student each.

You may also be interested:

• Divisibility criteria
• What is the rest of the division for?
• List of Least Common Multiple Exercises – MMC
• Greatest common divisor – GCD