I) find the smallest number by which 12748 be multiplied so that the product is a perfect square.
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The questions posted on the site are solely user generated, Doubtnut has no ownership or control over the nature and content of those questions. Doubtnut is not responsible for any discrepancies concerning the duplicity of content over those questions. Find the smallest number by which 12748 be multiplied so that the product is a perfect square.(a) 3187(b) 7814(c) 4577(d) 1842Answer Verified Hint: We arrange the numbers from given in the option in ascending order 1842, 3187, 4577, 7814. We see that the least number is 1842. We first assume $x=12748$ and $y=1842$ then multiply them and find the prime factorization of their product $xy$. If all the prime factors in prime factorization occur even number of times, then the product is a perfect square otherwise, we move to the next least numbers 3187, 4577, 7814. Complete step-by-step answer: Note: We need to note that square root is always a positive quantity, so we have rejected the negative values in the problem but when we solve equations like ${{x}^{2}}=a, a> 0$ we have to take both the values because we are finding the roots. A perfect cube is a number whose cube root is an integer. There are other primality tests like the Fermat primality test in higher classes. We can represent any prime number in the form of (6n+1) or (6n-1). So, to check if the number 3187 is prime, we can try to represent it in this form. So, we can either have (3188-1) or (3186+1), out of these, 3186 is divisible by 6. So, we can represent 3187 in the form (6n+1), hence it is a prime number. Welcome to Sarthaks eConnect: A unique platform where students can interact with teachers/experts/students to get solutions to their queries. Students (upto class 10+2) preparing for All Government Exams, CBSE Board Exam, ICSE Board Exam, State Board Exam, JEE (Mains+Advance) and NEET can ask questions from any subject and get quick answers by subject teachers/ experts/mentors/students. Categories
What is the perfect square of 12748?Hence, 12748 should be multiplied by 3187 to make it a perfect square. Which 12748 be multiplied so that the product is a perfect square?So, we will multiply it by 3187 to make a pair. 12748 × 3187 = 2 × 2 × 3187 × 3187. Hence, the smallest number by which it should be divided to make a perfect square is 3187. Which 2592 be multiplied so that the product is a perfect square?Hence, 2592 should be multiplied by 2 to make it a perfect square. What is the smallest number by which 4851 should be multiplied so that the product becomes a perfect square?Therefore, the smallest number by which 4851 must be multiplied so that the product becomes a perfect square is 11. What is the perfect square of 12748?Hence, 12748 should be multiplied by 3187 to make it a perfect square.
Which 12748 be multiplied so that the product is a perfect square?So, we will multiply it by 3187 to make a pair. 12748 × 3187 = 2 × 2 × 3187 × 3187. Hence, the smallest number by which it should be divided to make a perfect square is 3187.
Which 2592 be multiplied so that the product is a perfect square?Hence, 2592 should be multiplied by 2 to make it a perfect square.
What is the smallest number by which 2592 must be multiplied so that the product is a perfect cube?∴ The given number should be multiplied by 3187.
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