How Many 3-Digit Numbers Are There That Are Multiples of 5: A Comprehensive Guide
Understanding and identifying 3-digit numbers that are multiples of 5 is a fascinating exercise. This article will guide you through the process of finding how many such numbers exist, using an arithmetic sequence approach, and explain the significance of this mathematical concept.
Introduction to Multiples of 5
A number is considered a multiple of 5 if it ends in 0 or 5. This simple condition forms the basis of our exploration into the world of 3-digit multiples of 5.
Step-by-Step Breakdown
To find the number of 3-digit multiples of 5, we will follow a step-by-step approach.
Step 1: Identifying the Range of 3-Digit Numbers
The range of 3-digit numbers is from 100 to 999. We need to identify the smallest and largest 3-digit multiples of 5 within this range.
Smallest 3-Digit Multiple of 5: The smallest 3-digit number is 100, which is a multiple of 5. Largest 3-Digit Multiple of 5: The largest 3-digit number is 999. The largest multiple of 5 less than or equal to 999 is 995.Step 2: Identifying the Sequence of Multiples of 5
The 3-digit multiples of 5 form an arithmetic sequence where: First Term (a): 100 Common Difference (d): 5 Last Term (l): 995
Using the formula for the nth term of an arithmetic sequence:
[ l a (n-1) cdot d ]
We can plug in the known values:
[ 995 100 (n-1) cdot 5 ]
Subtracting 100 from both sides:
[ 895 (n-1) cdot 5 ]
Dividing both sides by 5:
[ 179 n - 1 ]
Adding 1 to both sides:
[ n 180 ]
Conclusion
There are 180 three-digit numbers that are multiples of 5.
Additional Insights
Let's explore an additional question that deals with 3-digit numbers that are both even and multiples of 5:
Three Darts in the Treble Twenty: 180
This question highlights the importance of precise language in mathematics. The term 'even number' in this context ties down the question nicely as even numbers that are multiples of 5 are also multiples of 10.
Given that the question specifies '3-digit' and assumes no leading zeroes, the first digit has nine choices (1-9) and the second digit has ten choices (0-9). The last digit must always be 0 for divisibility by 10.
The question does not preclude negative integers, so the correct answer would be double what some answers might think.
Conclusion
Understanding how to identify and derive 3-digit numbers that are multiples of 5 is not only mathematically rewarding but also useful in various real-world applications. Whether it's for coding, data analysis, or even in competitive mathematics, this knowledge remains invaluable.