There are 150 three-digit odd numbers that are divisible by 3.
Here's a breakdown of how to find that answer:
Understanding the Problem
- Three-digit numbers: These range from 100 to 999.
- Odd numbers: The last digit must be 1, 3, 5, 7, or 9.
- Divisible by 3: The sum of the digits must be divisible by 3.
Calculating the Solution
We can determine the number of 3-digit numbers divisible by 3. The reference states, "There are 300 3-digit numbers divisible by 3." However, we only want the odd ones.
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Total 3-digit numbers: There are 900 three-digit numbers (from 100 to 999).
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3-digit numbers divisible by 3: There are 300 three-digit numbers divisible by 3. (This is given information in the reference).
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Every other number divisible by 3 is odd: Roughly half of those numbers divisible by 3 will be odd. We can assume, therefore, that approximately half of these are odd, which is the actual case.
- This reasoning is based on the fact that there is a pattern: an even multiple of 3, an odd multiple of 3, and so on. So, when looking at numbers divisible by 3, every other number is odd.
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Calculation: Half of 300 = 150
Thus, there are 150 three-digit odd numbers divisible by 3.
Practical Insights
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Divisibility Rule of 3: A number is divisible by 3 if the sum of its digits is divisible by 3.
- Example: 102 is divisible by 3 because 1 + 0 + 2 = 3, and 3 is divisible by 3.
- Example: 105 is divisible by 3 because 1 + 0 + 5 = 6, and 6 is divisible by 3.
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Odd Number Identification: A number is odd if its last digit is 1, 3, 5, 7, or 9.
Summary
Combining both conditions, we determined that half of the total 3-digit numbers divisible by 3 are also odd. Therefore, there are 150 three-digit odd numbers that are divisible by 3.
