Q. 1
Quantitative Aptitude Test
Difficulty: Medium
(1 Mark)
All the prime factors of 182 are
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Step-by-Step Explanation & Concept Rationale
Prime factorization: 182=2×7×13.
Q. 2
Quantitative Aptitude Test
Difficulty: Medium
(1 Mark)
Least square number which contains 2100 as a factor?
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Step-by-Step Explanation & Concept Rationale
Find the prime factorization of 2100:
\(2100=21\times 100=(3\times 7)\times (2^{2}\times 5^{2})=2^{2}\times 3^{1}\times 5^{2}\times 7^{1}\)
Make it a perfect square:
For a number to be a perfect square, all the exponents in its prime factorization must be even numbers.
\(2^{2}\) and \(5^{2}\) already have even exponents.
\(3^{1}\) and \(7^{1}\) have odd exponents, so we must multiply by another 3 and 7 to make them even (\(3^{2}\) and \(7^{2}\)).
Calculate the least square number:
Multiply 2100 by the missing factors (\(3 \times 7 = 21\)):
\(2100\times 21=\mathbf{44100}\)
(Verification: \(\sqrt{44100} = 210\), which is a perfect integer).
\(2100=21\times 100=(3\times 7)\times (2^{2}\times 5^{2})=2^{2}\times 3^{1}\times 5^{2}\times 7^{1}\)
Make it a perfect square:
For a number to be a perfect square, all the exponents in its prime factorization must be even numbers.
\(2^{2}\) and \(5^{2}\) already have even exponents.
\(3^{1}\) and \(7^{1}\) have odd exponents, so we must multiply by another 3 and 7 to make them even (\(3^{2}\) and \(7^{2}\)).
Calculate the least square number:
Multiply 2100 by the missing factors (\(3 \times 7 = 21\)):
\(2100\times 21=\mathbf{44100}\)
(Verification: \(\sqrt{44100} = 210\), which is a perfect integer).
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