What are Multiples of 13?
When 13 is multiplied by any natural number, the result is known as a multiple of 13. The multiple of 13 can be written as 13n, where n can be any natural number, such as 1, 2, 3….
So, the multiples of 13 will be: 13, 26, 39, 52, 65, 78, 91, …
For example, if we multiply 13 by 3, the product is 39. As a result, we define 39 as a multiple of 13. Another thing to remember here is that the product of two numbers is a multiple of both the numbers. Hence, 39 is a multiple of 3 as well.
Also, we can observe that the difference between the consecutive multiples of 13, that is, the difference between each succeeding multiple of 13 and the preceding multiple of 13 is 13:
26 – 13 = 13,
39 – 26 = 13,
52 – 39 = 13 and so on.
For example, 65, 78 and 91 are all multiples of 13 because we can get these numbers by multiplying 13 by some natural number.
13 × 5 = 65 | 13 is multiplied by 5 to get 65 |
|---|---|
13 × 6 = 78 | 13 is multiplied by 6 to get 78 |
13 × 7 = 91 | 13 is multiplied by 7 to get 91 |
Finding the Multiples of 13 by Repeated Addition
Repeated addition means adding a number to itself multiple times. We can use repeated addition to find the multiple of a number.
If we want to find the first ten multiples of 13, we can write it in following way:
13 x 1 = 13 | 13 | |
|---|---|---|
13 x 2 = 26 | 13 + 13 = 26 | Here 13 is added two times |
13 x 3 = 39 | 13 + 13 + 13 = 39 | Here 13 is added three times |
13 x 4 = 52 | 13 + 13 + 13 + 13 = 52 | Here 13 is added four times |
13 x 5 = 65 | 13 + 13 + 13 + 13 + 13 = 65 | Here 13 is added five times |
13 x 6 = 78 | 13 + 13 + 13 + 13 + 13 + 13 = 78 | Here 13 is added six times |
13 x 7 = 91 | 13 + 13 + 13 + 13 + 13 + 13 + 13 = 91 | Here 13 is added seven times |
13 x 8 = 104 | 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 = 104 | Here 13 is added eight times |
13 x 9 = 117 | 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 = 117 | Here 13 is added nine times |
13 x 10 = 130 | 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 = 130 | Here 13 is added ten times |
The nth Multiple of 13
The nth multiple of 13 can be obtained by either multiplying 13 by n or by adding 13 n times. Here, n is any natural number such as 1, 2, 3, …
nth multiple of 13 = 13 n, or
nth multiple of 13 = 13 + 13 + 13 … n times
Solved Multiples of 13 Examples
Example 1: Is 65 a multiple of 13?
Solution:
List the multiples of 13: 13, 26, 39, 52, 65, 78, 91, …
65 is included in the above list.
Also, \(\frac{65}{13}\) = 5 R0. The remainder is 0 when 65 is divided by 13. So, 13 is a factor of 65.
Hence, 65 is a multiple of 13.
Example 2: What is the 9th multiple of 13?
Solution:
The 9th multiple of 13 can be obtained by the repeated addition of 13 for 9 times or multiplying 9 by 13.
13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 + 13 = 9 x 13 = 117
Hence, the 9th multiple of 13 is 117.
Example 3: What are the common multiples of 5 and 13?
Solution:
The multiples of 13 = 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, ….
The multiples of 5 = 5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, ….
We can see that 65 is common in the multiples of 13 and 5. So, every multiple of 65 will be a common multiple of 5 and 13.
The multiples of 65 = 65, 130, 195, 260, 325, …
So, the common multiples of 5 and 13 are the multiples of 65.
Example 4: John calculated the area of a rectangular park. Adrian asks John for the dimensions of the park, but John answers back in a riddle. John tells Adrian, “The park has an area of 104 square centimeters. Its width is 4 centimeters and its length is a multiple of 13.” Can you help Adrian find the length of the park?
Solution:
Area of the rectangular park = 104 square centimeters
Width, w = 4 centimeters
The length is stated as a multiple of 13. Any multiple of 13 can be written as 13n, where n is a natural number.
Hence, the length, l can be expressed as 13n.
Area of a rectangle A = l x w
104 = 13n x 4
n = \(\frac{104}{13~\times~4}\)
n = 2
The length l = 13n = 13 x 2 = 26 centimeters
Therefore, the length of the park is 26 centimeters.
The multiples of a number are obtained by multiplying a natural number by another natural number.
If a number is multiplied by another number, the resultant product is known as the multiple of the number, and those numbers that are multiplied are known as the factors of the resultant product.
Example : 13 x 7 = 91
91 is a multiple of both 7 and 13.
7 and 13 are factors of 91.
The multiples of 13 = 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, ….
So, the 4th multiple of 13 is 52.
To find the lowest common multiple of 3 and 13, let us first find the multiples of 3 and 13.
The multiples of 13 = 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, ….
The multiples of 3 = 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42 ….
We can see that 39 is the first number that is common in the multiples of both 3 and 13.
So, the least common multiple of 3 and 13 is 39.