Difference Between Factors and Multiples

Factors and multiples are two basic concepts in mathematics. Most of the time, people need clarification and to consider the difference between factors and multiples. They have distinct meanings and serve different purposes in different fields. In this article, we will see the difference between factors and multiples.

What are the Factors?

Any number that divides a particular number leaving no remains behind, is a Factor. For example, the factors of 32 are 1, 2, 4, 8, and 16 because they can divide 32 without any remainder. Here point to be noted is that factors always come in pairs. For example, if 4 is a factor of 12, 3 must also be a factor of 12 because 4 × 3 = 12. Similarly, if 6 is a factor of 12, then two must also be a factor of 12 because 6 × 2 = 12.

What are Multiples?

Multiples come when a number is multiplied by a whole number. For example, the multiples of 5 are 5, 10, 15, 20, 25, 30, 35, and so on because each of these numbers can be obtained by multiplying five by another whole number (1, 2, 3, 4, 5, 6, 7, and so on). Multiples can be infinite in number, and they become more extensive as we multiply the given number by more significant whole numbers.

Difference Between Factors and Multiples

Although factors and multiples are interrelated, they have some essential points of differences. The main difference between factors and multiples is the way they are calculated. Factors are obtained by dividing a given number by other numbers, whereas multiples are obtained by multiplying a given number. Hence, factors and multiples are obtained by division and multiplication, respectively.

Another difference between factors and multiples is their relationship to the original number. Factors are always smaller than or equal to the number provided, whereas multiples are always greater than or equal to the actual number. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12, which are all smaller than or equal to 12. On the other hand, the multiples of 12 include 12, 24, 36, 48, and so on, all greater than or equal to 12.

Factors are limited in number, whereas multiples are unlimited. For example, the factors of 12 are 1, 2, 3, 4, 6, and 12 only. But, the multiples of 12 are 12, 24, 36, 48, and so on.

Finally, factors and multiples are used in different areas of mathematics. Factors are used mainly in number theory and algebra, whereas multiples are used in arithmetic and geometry. In geometry, multiples are used to find the scale factor between two similar figures: the ratio of the corresponding sides.

Some Examples Based on Factors

Example 1: Find the greatest common factor (GCF) of 24 and 36.

Solution: To find the GCF of 24 and 36, we must find the largest number to divide 24 and 36 without leaving any remainder. First, we must find all the factors of both numbers and then detect the greatest common factor.

The factors of 24 include 1, 2, 3, 4, 6, 8, 12, and 24.

The factors of 36 have 1, 2, 3, 4, 6, 9, 12, 18, and 36.

The common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12.

Therefore, the GCF of 24 and 36 is 12.

Example 2: Find all the factors of 24.

Solution: To find the factors of 24, we need to find all the numbers that can divide 24 without leaving any remainder. The factors of 24 are 1, 2, 3, 4, 6, 8, 12, and 24.

We can find the factors of 24 by dividing 24 by each number from 1 to 24 and checking if the division leaves no remainder. For example, 24 ÷ 3 = 8, so 3 and 8 are factors of 24.

Example 3: Determine whether 15 is a factor of 75.

Solution: To determine whether 15 is a factor of 75, we need to check if 15 can divide 75 without leaving any remainder. We can do this by dividing 75 by 15. If the division leaves no remainder, then 15 is a factor of 75.

75 ÷ 15 = 5, which means that 15 is a factor of 75. We can also say that 75 is a multiple of 15 since 15 multiplied by 5 gives 75.

We can use the prime factorization or common factor method to find the greatest common factor (GCF) of 48 and 72. Here are both methods:

Method 1: Prime factorization method

Step 1: Find the prime factorization of 48 and 72.

Prime factorization of 48: 2 × 2 × 2 × 2 × 3 = 2^4 × 3

Prime factorization of 72: 2 × 2 × 2 × 3 × 3 = 2^3 × 3^2

Step 2: Identify the common prime factors and their smallest exponents.

The common prime factors of 48 and 72 are 2 and 3. The smallest exponent for 2 is 3, and for 3 is 1.

Step 3: Multiply the common prime factors raised to their smallest exponents to find the GCF.

GCF of 48 and 72 = 2^3 × 3^1 = 24

Therefore, the greatest common factor of 48 and 72 is 24.

Method 2: Common factor method

Step 1: List factors 48 and 72.

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Factors of 72: 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72

Step 2: Identify the common factors of 48 and 72.

The common factors of 48 and 72 are 1, 2, 3, 4, 6, 8, 12, and 24.

Step 3: Find the greatest common factor from the common factors.

The greatest common factor of 48 and 72 is 24.

Therefore, the greatest common factor of 48 and 72 is 24, the same as the answer we got using the prime factorization method.

Some Examples Based on Multiples

Example 1: List the first 5 multiples of 6.

Solution: To find the multiples of 6, we must multiply 6 by all the natural numbers. The first 5 multiples of 6 are:

6 × 1 = 6

6 × 2 = 12

6 × 3 = 18

6 × 4 = 24

6 × 5 = 30

Example 2: Determine whether 50 is a multiple of 3

Solution: No, 50 is not a multiple of 3.

A multiple of 3 is a number that can be evenly divided by 3. In other words, a number is a multiple of 3 if the remainder of the division by 3 is 0. When we divide 50 by 3, we get a remainder of 2. Therefore, 50 is not a multiple of 3.

Example 2: Determine whether 24 is a multiple of 3.

Solution: To determine whether 24 is a multiple of 3, we need to check if 3 can divide by 24 without leaving any remainder. We can do this by dividing 24 by 3. If the division leaves no remainder, 24 is a multiple of 3.

24 ÷ 3 = 8, meaning 24 is a multiple of 3. We can also say that 8 is the factor of 24, since 3 multiplied by 8 gives 24.

Example 3: Find the least common multiple (LCM) of 12 and 18.

Solution: To find the LCM of 12 and 18, we need to find the smallest number, a multiple of both 12 and 18. One way to do this is to list the multiples of both numbers until we find the first common multiple.

The multiples of 12 are 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, ...

The multiples of 18 are 18, 36, 54, 72, 90, 108, 126, 144, ...

The first common multiple of 12 and 18 is 36, so the LCM of 12 and 18 is 36.

We can also use the prime factorization method to find the LCM of 12 and 18. The prime factorization of 12 is 2 × 2 × 3, and the prime factorization of 18 is 2 × 3 × 3. To find the LCM, we take the product of all the prime factors and raise it to the greatest exponent. In this case, the prime factors are 2 and 3, and the greatest exponent for 2 is 2, and for 3 is 2. Therefore, the LCM of 12 and 18 is 2 × 2 × 3 × 3 = 36.

Conclusion

Factors and multiples are two essential topics in mathematics. Although they are related, they have different applications and meanings and serve different purposes in various fields. Factors are used primarily in number theory, whereas multiples are used in arithmetic and geometry. By understanding the difference between factors and multiples, we can better understand the properties of numbers and their relationships to each other.