Answered

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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 :

GinnyAnswer
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]
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