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112 MATHEMATICS

2 216 Observe that each prime factor of a number appears
three times in the prime factorisation of its cube. Do you remember that
2 108 a × b = (a × b) m
m
m
In the prime factorisation of any number, if each factor
2 54
appears three times, then, is the number a perfect cube?
3 27 Think about it. Is 216 a perfect cube?
3 9 By prime factorisation, 216 = 2 × 2 × 2 × 3 × 3 × 3
3 3 Each factor appears 3 times. 216 = 2 × 3 = (2 × 3) 3
3
3
1 =6 which is a perfect cube! factors can be
3
grouped in triples
Is 729 a perfect cube? 729 = 3 × 3 × 3 × 3 × 3 × 3
Yes, 729 is a perfect cube.
Now let us check for 500.
Prime factorisation of 500 is 2 × 2 × 5 × 5 × 5.

So, 500 is not a perfect cube.
Example 1: Is 243 a perfect cube? There are three
5’s in the product but
only two 2’s.
Solution: 243 = 3 × 3 × 3 × 3 × 3
In the above factorisation 3 × 3 remains after grouping the 3’s in triplets. Therefore, 243 is
not a perfect cube.

TRY THESE

Which of the following are perfect cubes?
1. 400 2. 3375 3. 8000 4. 15625
5. 9000 6. 6859 7. 2025 8. 10648

7.2.2 Smallest multiple that is a perfect cube
Raj made a cuboid of plasticine. Length, breadth and height of the cuboid are 15 cm,
30 cm, 15 cm respectively.
Anu asks how many such cuboids will she need to make a perfect cube? Can you tell?

Raj said, Volume of cuboid is 15 × 30 × 15 = 3 × 5 × 2 × 3 × 5 × 3 × 5
= 2 × 3 × 3 × 3 × 5 × 5 × 5
Since there is only one 2 in the prime factorisation. So we need 2 × 2, i.e., 4 to make
it a perfect cube. Therefore, we need 4 such cuboids to make a cube.
Example 2: Is 392 a perfect cube? If not, find the smallest natural number by which
392 must be multiplied so that the product is a perfect cube.
Solution: 392 = 2 × 2 × 2 × 7 × 7

The prime factor 7 does not appear in a group of three. Therefore, 392 is not a perfect
cube. To make its a cube, we need one more 7. In that case
392 × 7 = 2 × 2 × 2 × 7 × 7 × 7 = 2744 which is a perfect cube.