Find the least positive integer that should be multiplied to 720 so that the product obtained is a perfect square.

Answer

To find the least positive integer that should be multiplied to 720 so that the product obtained is a perfect square, we can follow these steps:

### Step 1: Factorize 720
First, we need to find the prime factorization of 720.

1. Divide 720 by 2 (the smallest prime number):
- \( 720 \div 2 = 360 \)
2. Divide 360 by 2:
- \( 360 \div 2 = 180 \)
3. Divide 180 by 2:
- \( 180 \div 2 = 90 \)
4. Divide 90 by 2:
- \( 90 \div 2 = 45 \)
5. Now, 45 is not divisible by 2, so we divide by 3:
- \( 45 \div 3 = 15 \)
6. Divide 15 by 3:
- \( 15 \div 3 = 5 \)
7. Finally, we are left with 5, which is a prime number.

Thus, the prime factorization of 720 is:
\[ 720 = 2^4 \times 3^2 \times 5^1 \]

### Step 2: Analyze the Exponents
For a number to be a perfect square, all the exponents in its prime factorization must be even.

- In \( 720 = 2^4 \times 3^2 \times 5^1 \):
- The exponent of 2 is 4 (even).
- The exponent of 3 is 2 (even).
- The exponent of 5 is 1 (odd).

### Step 3: Determine the Required Multiplication
Since the exponent of 5 is odd, we need to multiply by 5 to make it even.

### Step 4: Conclusion
The least positive integer that should be multiplied to 720 to make it a perfect square is:
\[ \text{Answer} = 5 \]

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