Q. 55.0( 3 Votes )
Let us calculate with which positive smallest number should be multiplied with the numbers below so that the product will be a perfect cube.
i. 675 ii. 200
iii. 108 iv. 121
v. 1225
Answer :
Let us recall what a perfect cube is.
A perfect cube is a number that is the cube of an integer.
(i). For 675.
We need to factorize 675. We get,
675 = 3 × 3 × 3 × 5 × 5
Group these factors into three similar integers. We have
675 = (3 × 3 × 3) × 5 × 5
To make it into a perfect cube, multiply both sides by 5.
We have
675 × 5 = (3 × 3 × 3) × 5 × 5 × 5
Now, group these factors into three similar integers.
3375 = (3 × 3 × 3) × (5 × 5 × 5)
We are able to group it in three similar integers. This means, 3375 is a perfect cube.
Thus, smallest positive number that should be multiplied with 675 is 5.
(ii). For 200.
We need to factorize 200. We get,
200 = 2 × 2 × 2 × 5 × 5
Group these factors into three similar integers. We have
200 = (2 × 2 × 2) × 5 × 5
To make it into perfect cube, multiply both sides by 5.
We have
200 × 5 = (2 × 2 × 2) × 5 × 5 × 5
Now, group these factors into three similar integers.
1000 = (2 × 2 × 2) × (5 × 5 × 5)
We are able to group it in three similar integers. This means, 1000 is a perfect cube.
Thus, smallest positive number that should be multiplied with 200 is 5.
(iii). For 108.
We need to factorize 108. We get,
108 = 2 × 2 × 3 × 3 × 3
Group these factors into three similar integers. We have
108 = (3 × 3 × 3) × 2 × 2
To make it into perfect cube, multiply both sides by 2.
We have
108 × 2 = (3 × 3 × 3) × 2 × 2 × 2
Now, group these factors into three similar integers.
216 = (3 × 3 × 3) × (2 × 2 × 2)
We are able to group it in three similar integers. This means, 216 is a perfect cube.
Thus, smallest positive number should be multiplied with 108 is 2.
(iv). For 121.
We need to factorize 121. We get,
121 = 11 × 11
We can’t group it into three similar integers. So, multiply both sides by 11.
121 × 11 = 11 × 11 × 11
Now, group these factors into three similar integers.
1331 = (11 × 11 × 11)
Since, we are able to group it in three similar integers. This means, 1331 is a perfect cube.
Thus, smallest positive number that should be multiplied with 121 is 11.
(v). For 1225.
We need to factorize 1225. We get,
1225 = 5 × 5 × 7 × 7
We can’t group it into three similar integers. So, multiply both sides by (5 × 7).
1225 × 5 × 7 = 5 × 5 × 7 × 7 × (5 × 7)
Now, group these factors into three similar integers.
42875 = (5 × 5 × 5) × (7 × 7 × 7)
Since, we are able to group it in three similar integers. This means, 42875 is a perfect cube.
Thus, smallest positive number that should be multiplied with 42875 is (5 × 7), i.e., 35.
Rate this question :


















Let us write the numbers given below as product of prime factors and write cube roots of the numbers:
i. 512 ii. 1728
iii. 5832 iv. 15625
v. 10648
West Bengal - Mathematics
Let us calculate by which positive smallest number should be divided with the numbers below so that the quotient will be a perfect cube.
i. 7000 ii. 2662
iii. 4394 iv. 6750
v. 675
West Bengal - Mathematics
Let us calculate with which positive smallest number should be multiplied with the numbers below so that the product will be a perfect cube.
i. 675 ii. 200
iii. 108 iv. 121
v. 1225
West Bengal - Mathematics
Let us write which number is not a perfect cube in the numbers given below.
i. 216 ii. 343
iii. 1024 iv. 324
v. 1744 vi. 1372
West Bengal - Mathematics
Let’s form two cubes with sides of length 5 cm and 1 cm
Let us calculate the number of small cube to form the big cube.
West Bengal - Mathematics
Debnath has made a cuboid whose length, breadth and height are 4 cm, 3 cm and 2 cm respectively. Let’s see how many cuboids of this type will form a cube.
West Bengal - MathematicsLet's find perfect cubes from the following numbers
125, 500, 64, 73, 729, 968
West Bengal - Mathematics