Find the smallest number by which 27783 be multiplied to get a perfect square number .

Answer

To find the smallest number by which 27783 must be multiplied to obtain a perfect square, we will follow these steps:

### Step 1: Factorization of 27783
First, we need to factorize the number 27783. We will check for divisibility by prime numbers.

- Since 27783 is odd, it is not divisible by 2.
- Next, we check divisibility by 3. We sum the digits: 2 + 7 + 7 + 8 + 3 = 27, which is divisible by 3.

Now we divide:
\[ 27783 \div 3 = 9261 \]

### Step 2: Continue Factorization
Next, we factor 9261. Again, we check for divisibility by 3:
- Sum of digits of 9261: 9 + 2 + 6 + 1 = 18, which is also divisible by 3.

Now we divide:
\[ 9261 \div 3 = 3087 \]

Continuing with 3087:
- Sum of digits: 3 + 0 + 8 + 7 = 18, divisible by 3.

Now we divide:
\[ 3087 \div 3 = 1029 \]

Continuing with 1029:
- Sum of digits: 1 + 0 + 2 + 9 = 12, divisible by 3.

Now we divide:
\[ 1029 \div 3 = 343 \]

### Step 3: Factor 343
Now we factor 343. We check for divisibility by 7:
\[ 343 \div 7 = 49 \]
And since \( 49 = 7 \times 7 \):
\[ 49 = 7^2 \]
So, \( 343 = 7^3 \).

### Step 4: Complete Factorization
Now we can write the complete factorization of 27783:
\[ 27783 = 3^4 \times 7^3 \]

### Step 5: Determine the Smallest Number to Multiply
For a number to be a perfect square, all the powers in its prime factorization must be even.

- The power of 3 is 4 (even).
- The power of 7 is 3 (odd).

To make the power of 7 even, we need to multiply by \( 7^1 \) (which is 7).

### Conclusion
Thus, the smallest number by which 27783 must be multiplied to obtain a perfect square is:
\[ \text{Answer} = 7 \]

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