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By which smallest number 90 must be multiplied so as to make it a perfect square ?
- a)10
- b)2
- c)5
- d)3
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
By which smallest number 90 must be multiplied so as to make it a perf...
The smallest to be multiplied with 90 so as to obtain a perfect square is 10.
90 * 10=900
square root of 900 is 30.
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By which smallest number 90 must be multiplied so as to make it a perf...
Introduction:
To find the smallest number by which 90 must be multiplied to make it a perfect square, we need to factorize 90 and find the smallest number that, when multiplied with 90, results in a perfect square.
Factorizing 90:
The prime factorization of 90 is as follows:
90 = 2 * 3^2 * 5
Finding the smallest number:
To make 90 a perfect square, we need to ensure that each prime factor has an even exponent. In other words, we need to pair up the prime factors.
Exponent of 2:
The exponent of 2 in the prime factorization of 90 is 1. To make it even, we need to multiply 90 by 2. Therefore, the exponent of 2 becomes 2.
Exponent of 3:
The exponent of 3 in the prime factorization of 90 is 2, which is already even.
Exponent of 5:
The exponent of 5 in the prime factorization of 90 is 1. To make it even, we need to multiply 90 by 5. Therefore, the exponent of 5 becomes 2.
Final factorization:
After multiplying 90 with 2 for the exponent of 2 and with 5 for the exponent of 5, we get:
90 * 2 * 5^2 = 90 * 2 * 25 = 4500
Verifying the result:
To check if 4500 is a perfect square, we can find its square root. The square root of 4500 is 67.0820393.
Conclusion:
Therefore, the smallest number by which 90 must be multiplied to make it a perfect square is 10 (option A).
To find the smallest number by which 90 must be multiplied to make it a perfect square, we need to factorize 90 and find the smallest number that, when multiplied with 90, results in a perfect square.
Factorizing 90:
The prime factorization of 90 is as follows:
90 = 2 * 3^2 * 5
Finding the smallest number:
To make 90 a perfect square, we need to ensure that each prime factor has an even exponent. In other words, we need to pair up the prime factors.
Exponent of 2:
The exponent of 2 in the prime factorization of 90 is 1. To make it even, we need to multiply 90 by 2. Therefore, the exponent of 2 becomes 2.
Exponent of 3:
The exponent of 3 in the prime factorization of 90 is 2, which is already even.
Exponent of 5:
The exponent of 5 in the prime factorization of 90 is 1. To make it even, we need to multiply 90 by 5. Therefore, the exponent of 5 becomes 2.
Final factorization:
After multiplying 90 with 2 for the exponent of 2 and with 5 for the exponent of 5, we get:
90 * 2 * 5^2 = 90 * 2 * 25 = 4500
Verifying the result:
To check if 4500 is a perfect square, we can find its square root. The square root of 4500 is 67.0820393.
Conclusion:
Therefore, the smallest number by which 90 must be multiplied to make it a perfect square is 10 (option A).
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