What is the smallest number by which 2880 must be divided to make it a perfect square?
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Join Examsbook 851 0 Q: What is the smallest number by which 2880 must be divided in order to make it into a perfect square ?
Answer : 4. "5" Explanation :Answer: D) 5 Explanation: By trial and error method, we get2880/3 = 960 is not a perfect square2880/4 = 720 is not a perfect square2880/5 = 576 which is perfect square of 24Hence, 5 is the least number by which 2880 must be divided in order to make it into a perfect square.Report ErrorPlease Enter Message Error Reported Successfully The Most Comprehensive Exam Preparation PlatformGet the Examsbook Prep App Today  Answer Verified Hint – In order to solve this problem first take the LCM of 2880 and then multiply the numbers which are not in pairs of three and then divide 2880 with the number obtained. Doing this will solve your problem and you will get the right answer. Complete step-by-step answer: $ \Rightarrow $2880=3×3×4×4×4×5 The primes 3 and 5 do not appear in groups of three. So, 2880 is not a perfect cube. In the factorization of 2880 the prime 3 appears only two times and the prime 5 appears once. So, if we divide 2880 by 3×5×5=75, then the prime factorisation of the quotient will not contain 3 and 5. Hence, the smallest natural number by which 2880 should be divided to make it a perfect cube is 75. Hence, the answer is 75. Note – In this problem you need to take the LCM of 2880 then you have to find the numbers which do not make a pair of three and divide 2880 by that number only. Be careful between cube and square root since sometimes we use the concept of square root instead of cube roots. Proceeding like this will solve this problem. To do : Nội dung chính Show
We have to find the numbers by which the given numbers must be multiplied so that the products are perfect squares and the numbers whose squares are the new numbers. Solution: Perfect Square: A perfect square has each distinct prime factor occurring an even number of times. (i) $8820=2\times2\times3\times3\times5\times7\times7$ $=(2)^2\times(3)^2\times5\times(7)^2$ $8820\times5=(2)^2\times(3)^2\times5\times(7)^2\times5$ $=(2\times3\times5\times7)^2$ $=(210)^2$ In order to make the pairs an even number of pairs, we have to multiply 8820 by 5, then the product will be the perfect square. Therefore, 5 is the smallest number by which 8820 must be multiplied so that the product is a perfect square and the number whose square is the new number is 210. (ii) $3675=3\times5\times5\times7\times7$ $=3\times(5)^2\times(7)^2$ $3675\times3=3\times(5)^2\times(7)^2\times3$ $=(3\times5\times7)^2$ $=(105)^2$ In order to make the pairs an even number of pairs, we have to multiply 3675 by 3, then the product will be the perfect square. Therefore, 3 is the smallest number by which 3675 must be multiplied so that the product is a perfect square and the number whose square is the new number is 105. (iii) $605=5\times11\times11$ $=5\times(11)^2$ $605\times5=5\times(11)^2\times5$ $=(5\times11)^2$ $=(55)^2$ In order to make the pairs an even number of pairs, we have to multiply 605 by 5, then the product will be the perfect square. Therefore, 5 is the smallest number by which 605 must be multiplied so that the product is a perfect square and the number whose square is the new number is 55. (iv) $2880=2\times2\times2\times2\times2\times2\times3\times3\times5$ $=(2)^2\times(2)^2\times(2)^2\times(3)^2\times5$ $2880\times5=(2)^2\times(2)^2\times(2)^2\times(3)^2\times(5)^2$ $=(2\times2\times2\times3\times5)^2$ $=(120)^2$ In order to make the pairs an even number of pairs, we have to multiply 2880 by 5, then the product will be the perfect square. Therefore, 5 is the smallest number by which 2880 must be multiplied so that the product is a perfect square and the number whose square is the new number is 120. (v) $4056=2\times2\times2\times3\times13\times13$ $=(2)^2\times2\times3\times(13)^2$ $4056\times2\times3=(2)^2\times2\times3\times(13)^2\times2\times3$ $=(2\times2\times3\times13)^2$ $=(156)^2$ In order to make the pairs an even number of pairs, we have to multiply 4056 by 6, then the product will be the perfect square. Therefore, 6 is the smallest number by which 4056 must be multiplied so that the product is a perfect square and the number whose square is the new number is 156. (vi) $3468=2\times2\times3\times17\times17$ $=(2)^2\times3\times(17)^2$ $3468\times3=(2)^2\times3\times(17)^2\times3$ $=(2\times3\times17)^2$ $=(102)^2$ In order to make the pairs an even number of pairs, we have to multiply 3468 by 3, then the product will be the perfect square. Therefore, 3 is the smallest number by which 3468 must be multiplied so that the product is a perfect square and the number whose square is the new number is 102. Updated on 10-Oct-2022 12:43:44
Solution : (i)By doing factorisation of `605` we get, What is the smallest number by which 2880 should be multiplied so that the result becomes a perfect square?∴ Hence, 5 is the smallest number by which 2880 must be divided in order to make it a perfect square. What should be multiplied to 3675 to make it a perfect square?For a number to be a perfect square, each prime factor has to be paired. Hence, 3675 must be multiplied by 3 for it to be a perfect square. What is the smallest number that should be multiplied to 1260 to make it a perfect square?The square root of 1260, (or root 1260), is the number which when multiplied by itself gives the product as 1260. Therefore, the square root of 1260 = √1260 = 6 √35 = 35.4964786985977. What should be multiplied to 288 to make it a perfect square?2 should be multiplied to 288 to become it a perfect square. How do you find the smallest number to divide by a perfect square?To get a perfect square, each factor of the given number must be paired. Hence, prime factor 7 does not have its pair. If the number is divided by 7, then the rest of the prime factor will be in pairs. Therefore, 252 has to be divided by 7 to get a perfect square.
What is the smallest number you need to multiply 280 by to make it a square number?This is Expert Verified Answer
So we should divide 280 by 2 × 7 × 5 = 70 to make it a perfect square. it should factors not HCF !!
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