AB denotes a two -digits number with digits as A and B. If `(AB)^(2) = AC C`, where ACC denotes a three - digits number having digits A,C and C. What is the value of A+ B + C if A,B,C are all distinct digits?

Options

A. 5
B. 6
C. 7
D. 9

Answer

To solve the problem, we need to find distinct digits A, B, and C such that the two-digit number AB squared equals the three-digit number ACC. Let's break it down step by step:

### Step 1: Understand the notation
- AB is a two-digit number where A is the tens digit and B is the units digit.
- ACC is a three-digit number where A is the hundreds digit and C is both the tens and units digit.

### Step 2: Set up the equation
We can express the two-digit number AB as:
\[ AB = 10A + B \]
The square of this number is:
\[ (10A + B)^2 = 100A^2 + 20AB + B^2 \]

The three-digit number ACC can be expressed as:
\[ ACC = 100A + 10C + C = 100A + 11C \]

### Step 3: Set the equation
From the problem statement, we have:
\[ (10A + B)^2 = 100A + 11C \]

### Step 4: Identify the range of A
Since AB is a two-digit number, A can range from 1 to 9 (as it cannot be 0).

### Step 5: Calculate the squares of two-digit numbers
We calculate the squares of two-digit numbers from 10 to 31 (since \(32^2 = 1024\) is a four-digit number):
- \(10^2 = 100\)
- \(11^2 = 121\)
- \(12^2 = 144\)
- \(13^2 = 169\)
- \(14^2 = 196\)
- \(15^2 = 225\)
- \(16^2 = 256\)
- \(17^2 = 289\)
- \(18^2 = 324\)
- \(19^2 = 361\)
- \(20^2 = 400\)
- \(21^2 = 441\)
- \(22^2 = 484\)
- \(23^2 = 529\)
- \(24^2 = 576\)
- \(25^2 = 625\)
- \(26^2 = 676\)
- \(27^2 = 729\)
- \(28^2 = 784\)
- \(29^2 = 841\)
- \(30^2 = 900\)
- \(31^2 = 961\)

### Step 6: Check for the form ACC
We need to find which of these squares is of the form ACC, where A, C, and B are distinct digits.

- **For 10:** \(100\) (digits: 1, 0, 0) - not distinct
- **For 11:** \(121\) (digits: 1, 2, 1) - not distinct
- **For 12:** \(144\) (digits: 1, 4, 4) - not distinct
- **For 13:** \(169\) (digits: 1, 6, 9) - distinct
- **For 14:** \(196\) (digits: 1, 9, 6) - distinct
- **For 15:** \(225\) (digits: 2, 2, 5) - not distinct
- **For 16:** \(256\) (digits: 2, 5, 6) - distinct
- **For 17:** \(289\) (digits: 2, 8, 9) - distinct
- **For 18:** \(324\) (digits: 3, 2, 4) - distinct
- **For 19:** \(361\) (digits: 3, 6, 1) - distinct
- **For 20:** \(400\) (digits: 4, 0, 0) - not distinct
- **For 21:** \(441\) (digits: 4, 4, 1) - not distinct
- **For 22:** \(484\) (digits: 4, 8, 4) - not distinct
- **For 23:** \(529\) (digits: 5, 2, 9) - distinct
- **For 24:** \(576\) (digits: 5, 7, 6) - distinct
- **For 25:** \(625\) (digits: 6, 2, 5) - distinct
- **For 26:** \(676\) (digits: 6, 7, 6) - not distinct
- **For 27:** \(729\) (digits: 7, 2, 9) - distinct
- **For 28:** \(784\) (digits: 7, 8, 4) - distinct
- **For 29:** \(841\) (digits: 8, 4, 1) - distinct
- **For 30:** \(900\) (digits: 9, 0, 0) - not distinct
- **For 31:** \(961\) (digits: 9, 6, 1) - distinct

### Step 7: Identify valid combinations
From the valid squares, we find:
- \(169\) corresponds to \(A = 1, B = 3, C = 9\)
- \(196\) corresponds to \(A = 1, B = 9, C = 6\)
- \(256\) corresponds to \(A = 2, B = 5, C = 6\)
- \(289\) corresponds to \(A = 2, B = 8, C = 9\)
- \(324\) corresponds to \(A = 3, B = 2, C = 4\) (valid)

### Step 8: Find distinct digits
The only valid combination with distinct digits is:
- For \(12^2 = 144\): \(A = 1, B = 2, C = 4\)

### Step 9: Calculate A + B + C
Now we calculate:
\[ A + B + C = 1 + 2 + 4 = 7 \]

### Final Answer
Thus, the value of \(A + B + C\) is \(7\).