Find the number which must be subtracted from 2509 to make it a perfect square.

Answer

To find the number that must be subtracted from 2509 to make it a perfect square, we can follow these steps:

### Step 1: Find the square root of 2509
We need to determine the nearest perfect square to 2509. To do this, we will find the square root of 2509 using the division method.

### Step 2: Pair the digits of 2509
We can pair the digits of 2509 from right to left. This gives us pairs: (25)(09).

### Step 3: Find the largest square less than or equal to 25
The largest perfect square less than or equal to 25 is 25 itself (since \(5^2 = 25\)).

### Step 4: Subtract and bring down the next pair
Subtract 25 from 25, which gives us 0. Bring down the next pair (09) next to 0, making it 09.

### Step 5: Double the quotient
Now, we double the current quotient (which is 5) to get 10.

### Step 6: Find the next digit
Now, we need to find a digit \(x\) such that \(10x \cdot x\) is less than or equal to 09. Testing \(x = 0\) gives us \(10 \cdot 0 \cdot 0 = 0\) (which is valid). Testing \(x = 1\) gives us \(10 \cdot 1 \cdot 1 = 10\) (which is too high). So, we take \(x = 0\).

### Step 7: Update the quotient
Now, our quotient is 50 (from 5 and 0).

### Step 8: Calculate the perfect square
The perfect square of 50 is \(50^2 = 2500\).

### Step 9: Subtract to find the number
Now, we subtract 2500 from 2509:
\[ 2509 - 2500 = 9 \]

### Conclusion
The number that must be subtracted from 2509 to make it a perfect square is **9**.

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