Discover the answers you need at Westonci.ca, where experts provide clear and concise information on various topics. Ask your questions and receive precise answers from experienced professionals across different disciplines. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

46. The least positive integer with which 3125 should be multiplied so that the product is a perfect square is [tex]\(\qquad\)[/tex]

(a) 4
(b) 5
(c) 6
(d) 2


Sagot :

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]