ABC + ABC + ABC = CCC. A, B, and C are unique numbers. Find CCC

Let's break down the equation: 

\[
ABC + ABC + ABC = CCC
\]

This can be rewritten as:

\[
3 \times ABC = CCC
\]

Where \(ABC\) is a three-digit number represented by \(100A + 10B + C\), and \(CCC\) is a number where all three digits are \(C\), meaning \(CCC = 111 \times C\).

Now, let's express the equation:

\[
3 \times (100A + 10B + C) = 111C
\]

Expanding both sides:

\[
300A + 30B + 3C = 111C
\]

Subtracting \(3C\) from both sides:

\[
300A + 30B = 108C
\]

Dividing the whole equation by 3 to simplify:

\[
100A + 10B = 36C
\]

Now, we need to find values for \(A\), \(B\), and \(C\) such that this equation holds, and \(A\), \(B\), and \(C\) are unique digits between 0 and 9.

Let's try different values of \(C\):

- For \(C = 1\), \(36C = 36\). But \(100A + 10B = 36\) has no valid digit combinations for \(A\) and \(B\).
- For \(C = 2\), \(36C = 72\). This gives \(100A + 10B = 72\), so \(A = 0\) and \(B = 7\).
  
Thus, \(ABC = 072\) and \(CCC = 222\).

Finally, let's check if this satisfies the original equation:

\[
072 + 072 + 072 = 222
\]

So, the solution is \(CCC = 222\).

Would you like more details or have any questions?

Here are some follow-up questions to deepen your understanding:
1. How would the equation change if \(CCC\) were \(333\)?
2. Can we find another solution if we allow \(A\), \(B\), or \(C\) to repeat?
3. Why does the equation work out for \(C = 2\) but not for higher values?
4. How would the problem differ if we changed the multiplier from 3 to 4?
5. What happens if \(A\), \(B\), and \(C\) were allowed to be non-unique?

**Tip:** When breaking down problems, isolating one variable at a time can simplify complex equations effectively.

ABC + ABC + ABC = CCC. A, B, and C are unique numbers. Find CCC.

Explore the solution to the equation ABC + ABC + ABC = CCC, where A, B, and C are unique digits. This problem involves algebraic manipulation and number theory concepts, and provides a step-by-step solution using linear equations. Suitable for Grades 8-10 students.

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