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Find the smallest number by which 180 must be multiplied so that the product becomes a perfect square.
- a)2
- b)3
- c)5
- d)10
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Find thesmallest numberby which180must be multiplied so that the produ...
180 = 22 × 32 × 5. To make all powers even, multiply by 5.
180 × 5 = 900 = 302
180 × 5 = 900 = 302
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Find thesmallest numberby which180must be multiplied so that the produ...
Understanding Perfect Squares
A perfect square is a number that can be expressed as the square of an integer. To determine the smallest number by which 180 must be multiplied to become a perfect square, we first need to factor 180 into its prime factors.
Prime Factorization of 180
- The prime factorization of 180 is:
- 180 = 2 × 2 × 3 × 3 × 5
- This can also be written as: 180 = 2^2 × 3^2 × 5^1
Analyzing the Prime Factors
- For a number to be a perfect square, all prime factors must have even exponents.
- In the prime factorization of 180:
- The exponent of 2 is 2 (even)
- The exponent of 3 is 2 (even)
- The exponent of 5 is 1 (odd)
Determining What is Needed
- The only prime factor with an odd exponent is 5.
- To make the exponent of 5 even, we need to multiply by 5.
- Therefore, we need to multiply 180 by 5 to ensure all exponents become even.
Conclusion
- The smallest number by which 180 must be multiplied to become a perfect square is:
- 5
Thus, the correct answer is option 'C'.
A perfect square is a number that can be expressed as the square of an integer. To determine the smallest number by which 180 must be multiplied to become a perfect square, we first need to factor 180 into its prime factors.
Prime Factorization of 180
- The prime factorization of 180 is:
- 180 = 2 × 2 × 3 × 3 × 5
- This can also be written as: 180 = 2^2 × 3^2 × 5^1
Analyzing the Prime Factors
- For a number to be a perfect square, all prime factors must have even exponents.
- In the prime factorization of 180:
- The exponent of 2 is 2 (even)
- The exponent of 3 is 2 (even)
- The exponent of 5 is 1 (odd)
Determining What is Needed
- The only prime factor with an odd exponent is 5.
- To make the exponent of 5 even, we need to multiply by 5.
- Therefore, we need to multiply 180 by 5 to ensure all exponents become even.
Conclusion
- The smallest number by which 180 must be multiplied to become a perfect square is:
- 5
Thus, the correct answer is option 'C'.
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Find thesmallest numberby which180must be multiplied so that the product becomes aperfect square.a)2b)3c)5d)10Correct answer is option 'C'. Can you explain this answer? for Class 8 2026 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about Find thesmallest numberby which180must be multiplied so that the product becomes aperfect square.a)2b)3c)5d)10Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 8 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find thesmallest numberby which180must be multiplied so that the product becomes aperfect square.a)2b)3c)5d)10Correct answer is option 'C'. Can you explain this answer?.
Find thesmallest numberby which180must be multiplied so that the product becomes aperfect square.a)2b)3c)5d)10Correct answer is option 'C'. Can you explain this answer? for Class 8 2026 is part of Class 8 preparation. The Question and answers have been prepared according to the Class 8 exam syllabus. Information about Find thesmallest numberby which180must be multiplied so that the product becomes aperfect square.a)2b)3c)5d)10Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 8 2026 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find thesmallest numberby which180must be multiplied so that the product becomes aperfect square.a)2b)3c)5d)10Correct answer is option 'C'. Can you explain this answer?.
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