Understanding Multiples: Counting Multiples of 3 and 5 Between 1 and 100
In this article, we will explore how to count the multiples of 3 and 5 between 1 and 100. We will also discuss finding the least common multiple (LCM) of two numbers and how to avoid double-counting certain numbers.
Introduction to Multiples
A multiple of a number is a product of that number and an integer. For example, the multiples of 3 are 3, 6, 9, 12, and so on, up to 99 within the range of 1 to 100.
Counting Multiples of 3 Between 1 and 100
Let's start with counting the multiples of 3 between 1 and 100. The smallest multiple of 3 in this range is 3, and the largest is 99.
Using the Arithmetic Sequence Formula
The number of terms in an arithmetic sequence can be found using the formula:
[text{Last term} text{First term} (n - 1) times text{Common difference} ]
Here, the common difference is 3. We can set up the equation to find the number of multiples of 3:
[99 3 (n - 1) times 3 ]
Solving for (n):
[99 3 3n - 3 ]
[99 3n ]
[33 n ]
Therefore, there are 33 multiples of 3 between 1 and 100.
That is: [3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, 51, 54, 57, 60, 63, 66, 69, 72, 75, 78, 81, 84, 87, 90, 93, 96, 99 ]
Counting Multiples of 5 Between 1 and 100
Now, let's count the multiples of 5 within the same range. The smallest multiple of 5 is 5, and the largest is 100.
Solving for Multiples of 5
We can use the same formula to find the number of multiples of 5:
[100 5 (n - 1) times 5 ]
Solving for (n):
[100 5 5n - 5 ]
[100 5n ]
[20 n ]
Therefore, there are 20 multiples of 5 between 1 and 100.
That is: [5, 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100 ]
Counting Multiples of 3 or 5 Between 1 and 100
When counting numbers that are multiples of either 3 or 5, we need to be careful not to double-count numbers that are multiples of both 3 and 5 (i.e., multiples of 15).
Least Common Multiple (LCM) of 3 and 5
The least common multiple (LCM) of 3 and 5 is 15. Therefore, we need to count the multiples of 15 as well:
Multiples of 15
The multiples of 15 between 1 and 100 are:
[15, 30, 45, 60, 75, 90 ]
There are 6 multiples of 15 in this range.
To find how many numbers are multiples of either 3 or 5, we use the following formula:
[33 (multiples of 3) 20 (multiples of 5) - 6 (multiples of 15) 47 ]
Therefore, there are 47 numbers between 1 and 100 that are either multiples of 3 or 5.
That is: [15, 30, 45, 60, 75, 90, 3, 6, 9, 12, 18, 21, 24, 27, 33, 36, 39, 42, 48, 51, 54, 57, 63, 66, 69, 72, 78, 81, 84, 87, 93, 96, 99, 5, 10, 20, 25, 35, 40, 50, 55, 65, 70, 80, 85, 95, 100, 3, 6, 9, 12 ]
Conclusion
By carefully considering the multiples of different numbers and their LCMs, we can accurately count and list numbers within a given range.
This article has shown you how to count multiples of 3, 5, and their LCM, 15, and provided examples to help solidify these concepts.