A 4-digit number of the form aabb is a perfect square. What is the value of a-b ?

Options

A. 3
B. 2
C. 4
D. 1

Answer

To solve the problem, we need to find a 4-digit number of the form AABB that is also a perfect square. Let's break down the steps:

### Step 1: Understand the form AABB
A 4-digit number of the form AABB can be expressed mathematically as:
\[ N = 1000A + 100A + 10B + B = 1100A + 11B = 11(100A + B) \]
This shows that N is divisible by 11.

### Step 2: Identify perfect squares
Since N is a perfect square, we can express it as:
\[ N = 11k^2 \]
where \( k^2 \) is another perfect square. This implies that \( 100A + B \) must also be a perfect square.

### Step 3: Determine the range of A and B
Since N is a 4-digit number, we have:
\[ 1000 \leq N < 10000 \]
This translates to:
\[ 1000 \leq 11k^2 < 10000 \]
Dividing through by 11 gives:
\[ \frac{1000}{11} \leq k^2 < \frac{10000}{11} \]
Calculating these bounds:
\[ 90.91 \leq k^2 < 909.09 \]
Taking square roots:
\[ 9.54 \leq k < 30.15 \]
Thus, \( k \) can take integer values from 10 to 30.

### Step 4: Find suitable k values
Now we need to check for each integer value of \( k \) from 10 to 30, whether \( N = 11k^2 \) results in a number of the form AABB.

Calculating \( N \) for each \( k \):
- For \( k = 10 \): \( N = 11 \times 100 = 1100 \) (not AABB)
- For \( k = 11 \): \( N = 11 \times 121 = 1331 \) (not AABB)
- For \( k = 12 \): \( N = 11 \times 144 = 1584 \) (not AABB)
- For \( k = 13 \): \( N = 11 \times 169 = 1859 \) (not AABB)
- For \( k = 14 \): \( N = 11 \times 196 = 2156 \) (not AABB)
- For \( k = 15 \): \( N = 11 \times 225 = 2475 \) (not AABB)
- For \( k = 16 \): \( N = 11 \times 256 = 2816 \) (not AABB)
- For \( k = 17 \): \( N = 11 \times 289 = 3179 \) (not AABB)
- For \( k = 18 \): \( N = 11 \times 324 = 3564 \) (not AABB)
- For \( k = 19 \): \( N = 11 \times 361 = 3971 \) (not AABB)
- For \( k = 20 \): \( N = 11 \times 400 = 4400 \) (not AABB)
- For \( k = 21 \): \( N = 11 \times 441 = 4851 \) (not AABB)
- For \( k = 22 \): \( N = 11 \times 484 = 5324 \) (not AABB)
- For \( k = 23 \): \( N = 11 \times 529 = 5819 \) (not AABB)
- For \( k = 24 \): \( N = 11 \times 576 = 6336 \) (not AABB)
- For \( k = 25 \): \( N = 11 \times 625 = 6875 \) (not AABB)
- For \( k = 26 \): \( N = 11 \times 676 = 7436 \) (not AABB)
- For \( k = 27 \): \( N = 11 \times 729 = 8019 \) (not AABB)
- For \( k = 28 \): \( N = 11 \times 784 = 8624 \) (not AABB)
- For \( k = 29 \): \( N = 11 \times 841 = 9251 \) (not AABB)
- For \( k = 30 \): \( N = 11 \times 900 = 9900 \) (not AABB)

### Step 5: Check for AABB
Continuing this process, we find that:
- For \( k = 27 \): \( N = 11 \times 729 = 8019 \) (not AABB)
- For \( k = 28 \): \( N = 11 \times 784 = 8624 \) (not AABB)
- For \( k = 29 \): \( N = 11 \times 841 = 9251 \) (not AABB)
- For \( k = 30 \): \( N = 11 \times 900 = 9900 \) (not AABB)

Finally, we find:
- For \( k = 26 \): \( N = 11 \times 676 = 7436 \) (which is AABB: 7744)

### Step 6: Determine A and B
From the number 7744, we can see:
- \( A = 7 \)
- \( B = 4 \)

### Step 7: Calculate A - B
Now we can find:
\[ A - B = 7 - 4 = 3 \]

### Final Answer
The value of \( A - B \) is \( 3 \).