Sure, let's solve this problem step-by-step.
### Step 1: Find the Prime Factorization of 1250
To determine which prime factor of 1250 is unpaired and the least number to be multiplied to 1250 to make it a perfect square, we start by finding the prime factorization of 1250.
The prime factorization of 1250 is:
- 1250 ÷ 2 = 625
- 625 ÷ 5 = 125
- 125 ÷ 5 = 25
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
So, the prime factorization of 1250 is:
[tex]\[ 1250 = 2^1 \times 5^4 \][/tex]
### Step 2: Identify the Unpaired Prime Factor
Next, we need to determine if any prime factors have odd exponents. Here, the prime factor 2 has an exponent of 1, and the prime factor 5 has an exponent of 4.
- The exponent of 2 (which is 1) is odd.
- The exponent of 5 (which is 4) is even.
Therefore, the unpaired prime factor is 2.
### Step 3: Determine the Least Number to Multiply 1250 to Make It a Perfect Square
A perfect square has all prime factors with even exponents. Since the exponent of 2 is odd (1), we need to multiply 1250 by 2 to make the exponent of 2 even.
So, the least number that needs to be multiplied to 1250 to make it a perfect square is 2.
### Conclusion
- The unpaired prime factor of 1250 is 2.
- The least number to be multiplied to 1250 to make it a perfect square is 2.
These steps lead us to our final answers:
1. The unpaired prime factor is [tex]\( 2 \)[/tex].
2. The least number to be multiplied to 1250 to make it a perfect square is [tex]\( 2 \)[/tex].
