Exercise 6.4
1. Find the cube of the following numbers
(i) 8
Solution:
(ii) 16
Solution:
(iii) 21
Solution:
(iv) 30
Solution:
2. Test whether the given numbers are perfect cubes or not.
(i) 243
Solution:
243 is a perfect cube:
243 =
Since the prime factors cannot be grouped into triplets, 243 is not a perfect cube.
(ii) 516
Solution:
516 is a perfect cube:
516 =
Since the prime factors cannot be grouped into triplets, 516 is not a perfect cube.
(iii) 729
Solution:
729 is a perfect cube:
729 = 3 × 3 × 3 × 3 × 3 × 3 =
Therefore, 729 is a perfect cube.
(iv) 8000
Solution:
8000 is a perfect cube:
8000 = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 =
Therefore, 8000 is a perfect cube.
(v) 2700
Solution:
2700 is a perfect cube:
2700 =
Since the prime factors cannot be grouped into triplets, 2700 is not a perfect cube.
3. Find the smallest number by which 8788 must be multiplied to obtain a perfect cube?
Solution:
Prime factorize 8788: 8788 =
To make it a perfect cube, we need one more
The smallest number to multiply is
4. What smallest number should 7803 be multiplied with so that the product becomes a perfect cube?
Solution:
Prime factorize 7803: 7803 =
Where, 27 =
=
To make it a perfect cube, we need one more
The smallest number to multiply is
5. Find the smallest number by which 8640 must be divided so that the quotient is a perfect cube?
Solution:
Prime factorize 8640: 8640 =
To make it a perfect cube by division, we need to remove
The smallest number to divide by is 2 × 5 =
6. Ravi made a cuboid of plasticine of dimensions 12cm, 8cm and 3cm. How many minimum number of such cuboids will be needed to form a cube?
Solution:
Volume of the cuboid = 12 × 8 × 3 =
Prime factorize the dimensions: 12 =
To form a cube, each dimension needs to be a perfect cube. The LCM of the powers needs to be a multiple of
We need one more
The minimum number of cuboids needed is 2 × 3 × 3 =
7. Find the smallest prime number dividing the sum 3¹¹ + 5¹³
Solution:
The sum of two odd numbers is
The smallest prime number that divides any even number is