Factors and Multiples Solved Examples

# Factors and Multiples Solved Examples | The Complete SAT Course - Class 10 PDF Download

## What are Factors and Multiples?

Factors and multiples are mathematical concepts that are closely related. A factor of a number is a number that can divide the given number exactly without leaving any remainder. In other words, a factor is a divisor of a given number. On the other hand, a multiple of a number is the product of that number and any other integer. In simpler terms, if you can obtain a particular number by multiplying two or more numbers, then those numbers are factors of that number, and the result is a multiple of those numbers.

For example, if we have the numbers 5 and 3, we can obtain the number 15 by multiplying them together, which means that 5 and 3 are factors of 15. Hence, 15 is a multiple of both 5 and 3. This relationship between factors and multiples is fundamental in mathematics and is used in many different contexts, including algebra and number theory.

### Factors and Multiples Questions with Solutions

Here are the steps to find the factors of a number:

• Choose two numbers that, when multiplied together, give the target number.
• Test whether these two numbers divide the target number without leaving a remainder.
• If they do, then they are factors of the target number.
• Write down the pair of factors.
• Repeat steps 1-4 with other number pairs until you have found all the factors.
• Remember that every number has at least two factors: 1 and the number itself.

For example, to find the factors of 35, we can start by trying 5 and 7 as the number pairs. We test whether they divide 35 exactly, and since they do, we know that 5 and 7 are factors of 35. Therefore, we can write 35 as the product of 5 and 7, i.e., 35 = 5 x 7.

1. What are the factors of 9?

The factors of 9 are the numbers that can divide 9 exactly without leaving any remainder. These numbers are 1, 3, and 9. We can see this by listing all the possible pairs of numbers that multiply to give 9:
1 × 9 = 9
3 × 3 = 9
9 × 1 = 9
Therefore, the factors of 9 are 1, 3, and 9.

2. What is the sum of factors of 12?

To find the sum of factors of 12, we need to first list all the factors of 12. The factors of 12 are 1, 2, 3, 4, 6, and 12.
To find the sum of these factors, we simply add them up:
1 + 2 + 3 + 4 + 6 + 12 = 28
Therefore, the sum of factors of 12 is 28.

3. Find the common factors of 30 and 45.

To find the common factors of 30 and 45, we need to list all the factors of each number and identify the factors that they have in common.
The factors of 30 are: 1, 2, 3, 5, 6, 10, 15, and 30.
The factors of 45 are: 1, 3, 5, 9, 15, and 45.
So, the common factors of 30 and 45 are 1, 3, 5, and 15. These are the numbers that can divide both 30 and 45 without leaving a remainder.

4.  What is the greatest common factor of 3 and 15.

To find the greatest common factor (GCF) of 3 and 15, we need to list all the factors of each number and identify the largest factor that they have in common.
The factors of 3 are: 1 and 3.
The factors of 15 are: 1, 3, 5, and 15.
So, the greatest common factor of 3 and 15 is 3, which is the largest number that can divide both 3 and 15 without leaving a remainder. Therefore, the GCF of 3 and 15 is 3.

5. Find the greatest common factor of 20 and 6?

To find the greatest common factor (GCF) of 20 and 6, we need to list all the factors of each number and identify the largest factor that they have in common.
The factors of 20 are: 1, 2, 4, 5, 10, and 20.
The factors of 6 are: 1, 2, 3, and 6.
The common factors of 20 and 6 are: 1 and 2.
Therefore, the greatest common factor of 20 and 6 is 2, which is the largest number that can divide both 20 and 6 without leaving a remainder.

### How to Find Multiples?

To find the multiples of a number, we can multiply the number by different whole numbers. The resulting numbers are known as the multiples of that number. For instance, if we take the number 35, then all the numbers obtained by multiplying 35 by another whole number are the multiples of 35, such as 35, 70, 105, 140, and so on. One of the simplest ways to find multiples is to use the skip counting method, where we add the number repeatedly to obtain its multiples.

6. Find the first 10 multiples of 10.

To find the first 10 multiples of 10, we can use the skip counting method and add 10 to the previous multiple to obtain the next one.
The first multiple of 10 is 10 itself.
To get the next multiple, we add 10 to 10, which gives us 20.
To get the third multiple, we add 10 to 20, which gives us 30.
We can continue this process to obtain the first 10 multiples of 10:
1 × 10 = 10
2 × 10 = 20
3 × 10 = 30
4 × 10 = 40
5 × 10 = 50
6 × 10 = 60
7 × 10 = 70
8 × 10 = 80
9 × 10 = 90
10 × 10 = 100
Therefore, the first 10 multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100.

7. What are the first five multiples of 7?

To find the first five multiples of 7, we can use the skip counting method and add 7 to the previous multiple to obtain the next one.
Therefore, the first five multiples of 7 are 7, 14, 21, 28, and 35.

8. Find the least common multiple of 12 and 18.

To find the least common multiple (LCM) of 12 and 18, we can use different methods such as listing out their multiples or using prime factorization. Here, we'll use the prime factorization method:
Step 1: Find the prime factorization of each number.
12 = 22 x 3
18 = 2 x 32
Step 2: Write down the prime factors with their highest powers:
22 x 32
Step 3: Multiply the factors:
LCM of 12 and 18 = 22 x 32 = 36
Therefore, the least common multiple of 12 and 18 is 36.

9. Find the common multiples of 3 and 5.

To find the common multiples of 3 and 5, we can use the skip counting method and look for the numbers that appear in both the multiples of 3 and 5.
The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ...
The multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, ...
We can see that the common multiples of 3 and 5 are 15, 30, 45, 60, ... and so on.
Therefore, the common multiples of 3 and 5 are any numbers that are multiples of both 3 and 5, such as 15, 30, 45, 60, and so on.

10. What are the first three common multiples 4 and 8?

To find the common multiples of 4 and 8, we can use the skip counting method and look for the numbers that appear in both the multiples of 4 and 8.
The multiples of 4 are: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ...
The multiples of 8 are: 8, 16, 24, 32, 40, ...
We can see that the common multiples of 4 and 8 are 8, 16, and 24.
Therefore, the first three common multiples of 4 and 8 are 8, 16, and 24.

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