Show that `1764` is a perfect square. Find the number whose square is `1764`.

Options

A. `42`
B. `44`
C. `38`
D. `48`

Explanation

To show that 1764 is a perfect square and to find the number whose square is 1764, we can use the long division method. Here’s a step-by-step solution:

### Step 1: Pair the digits
We start by pairing the digits of 1764 from the right. We have:
- 17 | 64 (the pairs are 17 and 64)

### Step 2: Find the largest square less than or equal to the first pair
Now, we look at the first pair, which is 17. The largest perfect square less than or equal to 17 is 3² = 9.
- Write down 3 and subtract 9 from 17.
- 17 - 9 = 8

### Step 3: Bring down the next pair
Next, we bring down the next pair of digits (64) next to the remainder (8), making it 864.

### Step 4: Double the quotient
Now, we double the quotient we have (which is 3). So, 2 * 3 = 6. We will use this to find the next digit.

### Step 5: Find the next digit
We need to find a digit (let's call it x) such that:
\[ (60 + x) \times x \leq 864 \]
Testing values for x:
- For x = 2:
\[ 62 \times 2 = 124 \] (too small)
- For x = 3:
\[ 63 \times 3 = 189 \] (too small)
- For x = 4:
\[ 64 \times 4 = 256 \] (too small)
- For x = 5:
\[ 65 \times 5 = 325 \] (too small)
- For x = 6:
\[ 66 \times 6 = 396 \] (too small)
- For x = 7:
\[ 67 \times 7 = 469 \] (too small)
- For x = 8:
\[ 68 \times 8 = 544 \] (too small)
- For x = 9:
\[ 69 \times 9 = 621 \] (too small)
- For x = 10:
\[ 70 \times 10 = 700 \] (too small)
- For x = 11:
\[ 71 \times 11 = 781 \] (too small)
- For x = 12:
\[ 72 \times 12 = 864 \] (just right)

So, we find that x = 12 works perfectly.

### Step 6: Write down the quotient
Now, we write down the quotient: 3 (from the first part) and 12 (from the second part), giving us 42.

### Conclusion
Thus, we have shown that 1764 is a perfect square, and the number whose square is 1764 is **42**.

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