3-Digit Number Puzzle: Sum Equals Product!

Nov 13, 2025 by Admin 43 views

Let's dive into a fun mathematical problem: finding the sum of all three-digit numbers where the sum of their digits is equal to the product of their digits. This is a cool challenge that combines basic arithmetic with a bit of logical thinking. So, grab your thinking caps, guys, and let's get started!

Understanding the Problem

Before we jump into solving, it's super important to really understand what we're trying to do. We're looking for 3-digit numbers, which means numbers from 100 to 999. For each of these numbers, we need to check if the sum of its individual digits is equal to the product of those same digits. If it is, we add that number to our running total. Finally, we need to provide that total sum. It sounds complex, but by breaking it down, it becomes manageable. Here's a quick example to illustrate the concept:

Let’s take the number 123. The sum of its digits is 1 + 2 + 3 = 6. The product of its digits is 1 * 2 * 3 = 6. Since the sum and the product are equal, 123 is one of the numbers we're looking for!

On the other hand, if we consider the number 234, the sum of the digits is 2 + 3 + 4 = 9, while the product is 2 * 3 * 4 = 24. Since 9 is not equal to 24, 234 does not meet our criteria.

To solve this problem effectively, we need to go through all the 3-digit numbers and test each one. We can accomplish this programmatically or manually, but either way, the method stays the same. Consider how each digit affects both the sum and the product; this will help you spot patterns or shortcuts to make the process faster. Remember, accuracy is key because each missed or wrongly included number will affect the final sum. So, double-check your work as you go!

Breaking Down the Solution

Okay, so how do we actually find these numbers and add them up? Here's a step-by-step approach that'll help you tackle this problem systematically. First, we need a way to go through all the 3-digit numbers. Think of it like this: we're checking each house on a street, one by one, to see if it meets our criteria. Second, for each number, we need to separate the digits. This means taking a number like 345 and figuring out that it's made up of 3, 4, and 5. Third, we add those digits together and then multiply them together. Finally, we compare the sum and the product. If they're the same, we add the original number to our total sum.

Let's put this into pseudocode:

sum = 0
for number from 100 to 999:
 a = hundreds digit of number
 b = tens digit of number
 c = units digit of number
 if a + b + c == a * b * c:
 sum = sum + number
print sum

This pseudocode helps to clarify the logical steps without getting bogged down in specific programming language syntax. To actually run this, you would need to convert it into a real programming language like Python, Java, or C++. When coding, always remember to test your solution with a few known cases to ensure accuracy before processing all 3-digit numbers. This testing phase is crucial for catching any logical errors or bugs in your code. It's better to find and fix mistakes early on than to get to the end and realize your total is incorrect.

Finding the Numbers

Now, let's roll up our sleeves and actually find those numbers! We're looking for 3-digit numbers where the sum of the digits equals the product of the digits. We can start by thinking about small digits. If we have a '0' in the number, the product will always be '0'. So, the sum of the digits must also be '0'. The only way this can happen is if all the digits are '0', but that's not a 3-digit number. So, no zeros allowed!

Let's try some combinations:

123: Sum = 1 + 2 + 3 = 6, Product = 1 * 2 * 3 = 6. This works!
132: Sum = 1 + 3 + 2 = 6, Product = 1 * 3 * 2 = 6. This works too!
213: Sum = 2 + 1 + 3 = 6, Product = 2 * 1 * 3 = 6. Another one!
231: Sum = 2 + 3 + 1 = 6, Product = 2 * 3 * 1 = 6. And another!
312: Sum = 3 + 1 + 2 = 6, Product = 3 * 1 * 2 = 6. Yet another!
321: Sum = 3 + 2 + 1 = 6, Product = 3 * 2 * 1 = 6. One more!

So, any permutation of 1, 2, and 3 works. That's six numbers already. Let's keep looking for other possibilities.

Are there any other combinations? Let's think about it. If we have larger digits, the product tends to grow much faster than the sum. So, we should focus on smaller digits. What about using '1' multiple times?

Let's consider the number 11x. The sum would be 1 + 1 + x = 2 + x, and the product would be 1 * 1 * x = x. So, we need 2 + x = x, which is never true. So, we can't have two '1's.

What if we have a '2'? Let's explore the possibility of the digits 2, 2, and x. Then the sum of the digits would be 2 + 2 + x = 4 + x and the product would be 2 * 2 * x = 4x. We want to find x such that 4 + x = 4x. Rearranging this equation, we get 3x = 4, which implies x = 4/3. Since x must be an integer, this doesn't give us any solutions.

Therefore, after some careful consideration, the only digits that satisfy the condition are 1, 2, and 3. This gives us the six permutations we already found: 123, 132, 213, 231, 312, and 321.

Calculating the Final Sum

Alright, we've identified the numbers: 123, 132, 213, 231, 312, and 321. Now, all that's left to do is add them up to find the final answer. This is the home stretch, guys!

So, here we go:

123 + 132 + 213 + 231 + 312 + 321 = ?

Let’s align them vertically to make addition easier:

Starting from the rightmost column (the ones place):

3 + 2 + 3 + 1 + 2 + 1 = 12. So, we write down '2' and carry over '1' to the next column.

Moving to the next column (the tens place):

2 + 3 + 1 + 3 + 1 + 2 + 1 (carried over) = 13. So, we write down '3' and carry over '1' to the next column.

Finally, the leftmost column (the hundreds place):

1 + 1 + 2 + 2 + 3 + 3 + 1 (carried over) = 13. So, we write down '13'.

Combining these results, we get 1332. Therefore, the sum of all 3-digit numbers where the sum of their digits equals the product of their digits is 1332.

So, the final answer is 1332!

Verification and Conclusion

To be absolutely sure, it’s always a good idea to double-check our work. We can use a calculator or a simple script to verify that the sum of 123, 132, 213, 231, 312, and 321 indeed equals 1332. Trust, but verify, as they say!

In conclusion, we’ve successfully solved the problem by systematically identifying all 3-digit numbers that meet the given condition and then summing them up. This exercise not only tests our arithmetic skills but also our logical thinking and problem-solving abilities. Keep practicing such problems to sharpen your math skills and boost your confidence.

So, there you have it! A fun mathematical puzzle solved step by step. I hope you enjoyed the process as much as I did. Keep exploring, keep learning, and keep those numbers crunching!