Unlocking the Mystery: How Many 3-Digit Numbers Can Be Divided Evenly by 12?
Have you ever wondered about the mathematical patterns that surround numbers? Today, we're diving into a common question that pops up: how many 3-digit numbers are divisible by 12? It might sound like a complex problem, but with a little explanation, it becomes quite straightforward for anyone to understand. Let's break it down!
Understanding Divisibility by 12
First things first, what does it mean for a number to be divisible by 12? It means that when you divide that number by 12, there's no remainder – it divides evenly. For a number to be divisible by 12, it must be divisible by both 3 and 4, because 3 and 4 are factors of 12, and they don't share any common factors other than 1.
Identifying the Range of 3-Digit Numbers
The "3-digit numbers" are the whole numbers that start at 100 and go all the way up to 999. So, our search is confined to this specific range.
Finding the First 3-Digit Number Divisible by 12
To find the first 3-digit number that's divisible by 12, we can start checking from 100. We can divide 100 by 12, which gives us 8 with a remainder of 4. This means 100 isn't divisible by 12. The next multiple of 12 after 8 x 12 (which is 96) is 9 x 12, which equals 108. Since 108 is a 3-digit number, it's our starting point!
So, the first 3-digit number divisible by 12 is 108.
Finding the Last 3-Digit Number Divisible by 12
Now, let's find the largest 3-digit number that's divisible by 12. The largest 3-digit number is 999. Let's divide 999 by 12. When we do this, we get 83 with a remainder of 3. This tells us that 999 is not perfectly divisible by 12. To find the closest number below 999 that *is* divisible by 12, we subtract the remainder from 999. So, 999 - 3 = 996.
Therefore, the last 3-digit number divisible by 12 is 996.
Calculating the Total Count
Now that we know our starting point (108) and our ending point (996), and we know that each of these numbers is a multiple of 12, we can figure out how many there are. This is a lot like counting items in a sequence where each item is a certain distance apart.
We can think of this as an arithmetic sequence. The formula for the number of terms in an arithmetic sequence is:
Number of terms = ((Last term - First term) / Common difference) + 1
In our case:
- The First term is 108.
- The Last term is 996.
- The Common difference is 12 (since we're looking for multiples of 12).
Let's plug these values into the formula:
Number of 3-digit numbers divisible by 12 = ((996 - 108) / 12) + 1
First, calculate the difference: 996 - 108 = 888.
Next, divide the difference by 12: 888 / 12 = 74.
Finally, add 1: 74 + 1 = 75.
So, there are 75 three-digit numbers that are divisible by 12.
A Different Approach: Using Division of the Range
Another way to think about this is to consider how many multiples of 12 are there up to 999, and then subtract how many multiples of 12 are there up to 99 (since we don't want 1-digit or 2-digit numbers).
Number of multiples of 12 up to 999: 999 / 12 = 83.25. We take the whole number part, which is 83. This means there are 83 multiples of 12 from 1 up to 999.
Number of multiples of 12 up to 99: 99 / 12 = 8.25. We take the whole number part, which is 8. This means there are 8 multiples of 12 from 1 up to 99.
To find the number of 3-digit multiples, we subtract the second number from the first:
83 - 8 = 75.
Both methods confirm our answer: there are 75 three-digit numbers divisible by 12.
Conclusion
Understanding how to find the number of multiples of a specific number within a given range is a fundamental concept in mathematics. Whether you're a student tackling homework or just curious about numbers, knowing these techniques can be very helpful. So, the next time someone asks how many 3-digit numbers are divisible by 12, you'll have the answer and the explanation ready!
Frequently Asked Questions (FAQ)
How do you find the first 3-digit number divisible by 12?
To find the first 3-digit number divisible by 12, you start with the smallest 3-digit number (100) and divide it by 12. If there's a remainder, you keep adding 12 to the result of the division (or just find the next multiple of 12) until you get a number that is 100 or greater. In this case, 108 is the first 3-digit number perfectly divisible by 12.
Why is it important to find the first and last numbers in the range?
Finding the first and last numbers that fit your criteria (in this case, being a 3-digit number divisible by 12) is crucial for calculating the total count. These numbers act as the boundaries of your sequence, allowing you to use mathematical formulas to determine how many numbers fall within that specific range and meet the divisibility requirement.
What is the quickest way to solve this type of problem?
The quickest way involves dividing the upper limit of the range (999) by 12 and taking the whole number, then dividing the number just below the lower limit of the range (99) by 12 and taking the whole number, and finally subtracting the second result from the first. This method directly gives you the count of multiples within the specified range.
