Find the smallest natural number that we need to divide 40 with to make it a perfect cube.
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Is $$250$$ a perfect cube? If not, then by which smallest natural number should $$250$$ be divided so that the quotient is a perfect cube? Show
Solution $$250 = 2\times \underbrace {5\times 5\times 5}$$ The prime factor $$2$$ does not appear in triplet. Therefore $$250$$ is not a perfect cube. Since in the Factorization, $$2$$ appears only one time. It we divide the number $$250$$ by $$2$$, then the quotient will not contain $$2$$. Rest can be expressed in cubes. $$\therefore 250\div 2 = 125$$ $$= 5\times 5\times 5 = 5^{3}$$. $$\therefore$$ The smallest number by which $$250$$ should be divided to make it a perfect cube is $$2$$. (i) 24 Resolving 24 into Prime factors as follows Here, factor 3 is not in group of three so 24 is not a perfect cube. Therefore, the smallest number by which us should be divided to make it a perfect cube is 3. (ii) 250 Resolving 250 into Prime factors as follows Here, factor 2 is not in group of three. ∴ The smallest number by which 250 should be divided to make it a perfect cube is 2. Factors of 250 = 5 × 5 × 5 × 2 5 can be grouped easily, thus we should divide 250 by 2 to get a perfect cube. Hope it helps you!
View Full Answer Prime factors of 250 are 2 x 5 x 5 x 5. 2 must be divided by 250 to get a perfect cube.
IS 250 is a perfect cube?Is 250 a Perfect Cube? The number 250 on prime factorization gives 2 × 5 × 5 × 5. Here, the prime factor 2 is not in the power of 3. Therefore the cube root of 250 is irrational, hence 250 is not a perfect cube. Is the smallest number by which 36 must be multiplied to obtain a perfect cube?∴ The least number which should be multiplied by 36 to make a cube is 6. IS 512 a perfect cube?Yes, 512 is a perfect cube number since the cube root value of 512 is a whole number. How do you cube 12?The cube root of 12 is expressed as ∛12 in the radical form and as (12)⅓ or (12)0.33 in the exponent form. ... Cube root of 12 in Radical Form: ∛12.. Download PDF
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What are Perfect Numbers?It is said that the Egyptians were the ones who discovered the perfect number. However, it was the curiosity of the Greeks which made them study the concept of the perfect number and bring it forward for the world to study. So, perfect numbers are the positive integers that are equal to the sum of their factors except for the number itself. In other words, perfect numbers are the positive integers which are the sum of their proper divisors. For example - The smallest perfect number is 6, which is the sum of its proper divisors: 1, 2 and 3 What is a Square?A square is a flat shape with four equal sides having a 900 angle in every interior. (Image will be Uploaded Soon) Perfect SquareA square of a number is acquired by multiplying a specific number to itself. We can say that an ideal square is a number that is obtained by the increase of a number without any other number. For instance, 4 = 2 x 2 25 = 5 x 5 Properties of Perfect Squares are:
What is a Cube?A Cube is a three-dimensional figure whose edges are of the same length. The volume of a cuboid is given by the product of its measurements. (Image will be Uploaded Soon) Perfect CubeThe idea of a Perfect Cube is a build-up to the idea of a perfect square. The thing that matters is that in cubes, we increase a number thrice. The perfect cube of a number can be characterized as the number obtained by multiplying a number by itself two times. For example, 8 is known as a perfect cube number because it is obtained by multiplying 2 thrice; i.e. 8 = 2 x 2 x 2. We may consider a perfect cube as a cube of each side measuring x units and the volume of such a cube will be x3 cubic units. Numbers like 23 = 8, 33 = 27, 43 = 64, then the numbers 8, 27, 64 are termed as perfect cubes. Therefore, a number is defined as a perfect cube, if it is a cube of another natural number. Suppose that a and b are two different natural numbers, then a is said to be a perfect cube of b if a = b x b x b i.e. a = b3 Let’s go ahead and understand more about perfect cubes. Cubes are an Important Part of Algebra.Algebra is the section of mathematics that deals with symbols and rules for control of these symbols. In elementary algebra, the symbols are known as variables that do not have fixed values. These variables form equations that then describe the relationship between them. Uses of Algebra
History of CubesCalculation of the cubes of large numbers was very common in many ancient civilizations. By the old Babylonian period (20th to 16th centuries BC), the Mesopotamian Mathematicians invented cuneiform tablets with tables for calculating cubes and cube roots. The ancient Greek Diophantus also knew about the Cubic equations. The hero of Alexandria designed a method for calculating cube roots in the 1st Century. The methods for solving cubic equations and
calculating cube roots are mentioned in the nine Chapters of Mathematical Art. Our own respected mathematician Aryabhata wrote an Explanation of cubes. In 2010 Alberto Zanoni found a faster squaring and multiplying way to calculate the cube of a long integer in a certain range. Properties of Perfect Cubes
For instance, 43 = 4 x 4 x 4 = 64 23 = 2 x 2 x 2 = 8
For instance, 33 = 3 x 3 x 3 = 27 73 = 7 x 7 x 7 = 343
For instance, 103 = 1000 ends in 0 113 = 1331 ends in 1 143 = 2744 ends in 4 53 = 125 ends in 5 163 = 4096 ends in 6 193 = 6859 ends in 9
13+23+33+43…+n3 = (1+2+3+…+n)2 Have a look at following example: 13+23+33 = 1 + 8 + 27 = 36 and (1+2+3)2 = (6)2 = 36 How to find whether a number is a perfect cube or not?To find that whether a number is a perfect square or not follow the steps mentioned below:
Examples of Perfect Cube Concept.
Solution: By factoring 512 into prime factors, we get 512 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 After grouping the numbers in triplets, we get 512 = (2 x 2 x 2) x (2 x 2 x 2) x (2 x 2 x 2) Hence 512 is a perfect cube as no number is leftover. Taking one number from each group, 512 = (2)x(2)x(2) = 8 Therefore, 512 is the perfect cube of 8.
Solution: After factoring 9261 into prime factors, we get 9261 = 3 x 3 x 3 x 7 x 7 x 7 Group these numbers into triplets 9261 = (3 x 3 x 3) x (7 x 7 x 7) No number is leftover, so 9261 is a perfect cube. Taking one number from each group, 9261 = (3) x (7) = 21 Hence, 9261 is the perfect cube of 21.
Solution: Factoring 1323 into prime factors, we get 1323 = 3 x 3 x 3 x 7 x 7 Now to group these factors into triplets, one more 7 is needed. Hence, 7 is the smallest number which should be multiplied to 1323 to make it a perfect cube.
Solution: We have 1375 = 11 x 5 x 5 x 5 1375 = 11 x (5 x 5 x 5) 11 is leftover which means that if we divide 1375 by 11, then the quotient will be a perfect cube. Dividing 1375 by 11, we get 137511/11 = 5×5×5 = 125 The number is 125 and 5 is its cube root. Perfect Cubes of First 50 Natural Numbers are as Follows:
12. 1728 = 12 × 12 × 12 = 123 13. 2197 = 13 × 13 × 13 = 133 14. 2744 = 14 × 14 × 14 = 143 15. 3375 = 15 × 15 × 15 = 153 16. 4096 = 16 × 16 × 16 = 163 17. 4913 = 17 × 17 × 17 = 173 18. 5832 = 18 × 18 × 18 = 183 19. 6859 = 19 × 19 × 19 = 193 20. 8000 = 20 × 20 × 20 = 203 21. 9261 = 21 × 21 × 21 = 213 22. 10648 = 22 × 22 × 22 = 223 23. 12167 = 23 × 23 × 23 = 233 24. 13824 = 24 × 24 × 24 = 243 25. 15625 = 25 × 25 × 25 = 253 26. 17576 = 26 × 26 × 26 = 263 27. 19683 = 27 × 27 × 27 = 273 28. 21952 = 28 × 28 × 28 = 283 29. 24389 = 29 × 29 × 29 = 293 30. 27000 = 30 × 30 × 30 = 303 31. 29791 = 31 × 31 × 31 = 313 32. 32768 = 32 × 32 × 32 = 323 33. 35937 = 33 × 33 × 33 = 333 34. 39304 = 34 × 34 × 34 = 343 35. 42875 = 35 × 35 × 35 = 353 36. 46656 = 36 × 36 × 36 = 363 37. 50653 = 37 × 37 × 37 = 373 38. 54872 = 38 × 38 × 38 = 383 39. 59319 = 39 × 39 × 39 = 393 40. 64000 = 40 × 40 × 40 = 403 41. 68921 = 41 × 41 × 41 = 413 42. 74088 = 42 × 42 × 42 = 423 43. 79507 = 43 × 43 × 43 = 433 44. 85184 = 44 × 44 × 44 = 443 45. 91125 = 45 × 45 × 45 = 453 46. 97336 = 46 × 46 × 46 = 463 47. 103823 = 47 × 47 × 47 = 473 48. 110592 = 48 × 48 × 48 = 483 49. 117649 = 49 × 49 × 49 = 493 50. 125000 = 50 × 50 × 50 = 503 SQUARE ROOTSA square root is a number which gives a specified quantity when multiplied by itself. The square of the number 3 is 9 and the square root of 9 is 3. Square and square root are inversely related. Method to find the square root of a number is as follows: To find the square root of a number, find some number that when multiplied by itself gives the original number. For example, to find the square root of 9, find the number that when multiplied by itself gives 9, which is 3, because 3 x 3 = 9. (Image will be Uploaded Soon) CUBE ROOTSA cube root is the number that multiplies by itself 3 times to create a cubic value. Cube roots like the perfect cube equations were created and used for calculation by the Babylonian mathematicians. The cube of the number 2 is 8 and the cube root of 8 is 2. Cubes and cube roots are inversely related. Methods to find the cube root of a number are as follows: In order to find the cube root of a number, find some number that when multiplied by itself twice gives you the original number. For instance, to find the cube root of 8, find the number that when multiplied by itself twice gives 8, which is 2, because 2 x 2 x 2 = 8. Key Points From the Chapter -
What should be divided by 53 to 40 to make it a perfect cube?∴ The smallest number by which 53240 should be divided to make it a perfect cube is 5. Was this answer helpful?
How do you find the smallest perfect cube?Naive Approach: Check all perfect cubes one by one starting from 1 and select the one which is divisible by all the elements of the array. Efficient Approach: Find the least common multiple of all the elements of the array and store it in a variable lcm. Find all prime factor of the found LCM.
What number should be divided to get a perfect cube?A number is a perfect cube only when each factor in the prime factorization is grouped in triples. Using this concept, the smallest number can be identified. Here, the prime factor 3 is not grouped as a triplet. Hence, we divide 81 by 3, so that the obtained number becomes a perfect cube.
What is the smallest number by which 704 should be divided to get a perfect cube?Therefore, 704 must be divided by 11 to make it a perfect cube.
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