Answer :
To determine the least positive integer with which 3125 should be multiplied so that the product is a perfect square, we follow these steps:
1. Prime Factorization of 3125:
First, we need to find the prime factors of 3125.
[tex]\[ 3125 \div 5 = 625 \\ 625 \div 5 = 125 \\ 125 \div 5 = 25 \\ 25 \div 5 = 5 \\ 5 \div 5 = 1 \][/tex]
Therefore, the prime factorization of 3125 is:
[tex]\[ 3125 = 5^5 \][/tex]
2. Checking the exponents in the prime factorization:
To make a number a perfect square, all the exponents in its prime factorization must be even. In our case, the exponent of the prime factor 5 is 5, which is odd.
3. Determining the required multiplier:
Since the exponent of 5 in the factorization of 3125 is odd, we need to multiply 3125 by 5 to make the exponent even (5 + 1 = 6), because 6 is the next even number after 5.
Mathematically:
[tex]\[ 3125 \times 5 = 5^5 \times 5 = 5^6 \][/tex]
5^6 is a perfect square because the exponent 6 is even.
4. Conclusion:
Therefore, the least positive integer with which 3125 should be multiplied to make the product a perfect square is:
[tex]\[ \boxed{5} \][/tex]
Hence, the correct answer is:
[tex]\[ \qquad \boxed{5} \][/tex]
1. Prime Factorization of 3125:
First, we need to find the prime factors of 3125.
[tex]\[ 3125 \div 5 = 625 \\ 625 \div 5 = 125 \\ 125 \div 5 = 25 \\ 25 \div 5 = 5 \\ 5 \div 5 = 1 \][/tex]
Therefore, the prime factorization of 3125 is:
[tex]\[ 3125 = 5^5 \][/tex]
2. Checking the exponents in the prime factorization:
To make a number a perfect square, all the exponents in its prime factorization must be even. In our case, the exponent of the prime factor 5 is 5, which is odd.
3. Determining the required multiplier:
Since the exponent of 5 in the factorization of 3125 is odd, we need to multiply 3125 by 5 to make the exponent even (5 + 1 = 6), because 6 is the next even number after 5.
Mathematically:
[tex]\[ 3125 \times 5 = 5^5 \times 5 = 5^6 \][/tex]
5^6 is a perfect square because the exponent 6 is even.
4. Conclusion:
Therefore, the least positive integer with which 3125 should be multiplied to make the product a perfect square is:
[tex]\[ \boxed{5} \][/tex]
Hence, the correct answer is:
[tex]\[ \qquad \boxed{5} \][/tex]
