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Example 4: Find the smallest natural number by which 2000 should be multiplied to make it a
perfect cube.
2 2000
Solution: Since, prime factorisation of 2000 = 2 × 2 × 2 × 2 × 5 × 5 × 5 2 1000
Here, prime factors 2 and 5 appear in the group of three, but one prime factor 2 is left 2 500
ungrouped, so if we multiply 2000 by (2 × 2), we will get one more triplet of 2. 2 250
5 125
∴ 2000 × (2 × 2) = 2 × 2 × 2 × 2 × 2 × 2 × 5 × 5 × 5 = 8000, which is a perfect cube. 5 25

Therefore, 2 × 2 = 4 is the smallest natural number by which 2000 must be multiplied 5 5
to make it perfect cube. 1
Example 5: Find the smallest natural number by which 3000 should be divided to make it a perfect cube.

Solution: Prime factorisation of 3000 = 2 × 2 × 2 × 3 × 5 × 5 × 5 2 3000
2 1500
Here, prime factors 2 and 5 appear in the group of three, but prime factor 3 is left 2 750
ungrouped, so if we divide 3000 by 3, we are left with a triplet of prime factors 2 and 3 375
5, making the quotient (3000 ÷ 3 = 1000) a perfect cube. Therefore, 3 is the smallest 5 125
natural number by which 3000 must be divided by 3 to make it perfect cube. 5 25
5 5
Properties of Perfect Cubes (Cube Numbers) 1

Let us observe the cubes of the first 20 natural numbers.

Natural number 1 2 3 4 5 6 7 8 9 10
Cube 1 8 27 64 125 216 343 512 729 1000
Natural number 11 12 13 14 15 16 17 18 19 20
Cube 1331 1728 2197 2744 3375 4096 4913 5832 6859 8000

In this table, we observe that the cubes of natural number possess the following properties:

Property 1

The cubes of all odd numbers are odd, and the cubes of all even numbers are even.
3
For example, 1 = 1 × 1 × 1 = 1 (odd) 2 = 2 × 2 × 2 = 8 (even)
3
3
3
5 = 5 × 5 × 5 = 125 (odd) 6 = 6 × 6 × 6 = 216 (even)
Property 2
The number of zeros at the end of a perfect cube is always a multiple of 3.

3
For example, 10 = 10 × 10 × 10 = 1000 (3 zeros)
3
500 = 500 × 500 × 500 = 125000000 (6 zeros)
3
2000 = 2000 × 2000 × 2000 = 8000000000 (9 zeros)
Property 3

The cube of the numbers having digits 1, 4, 5, 6 and 9 at its ones place are number ending in the
same digits respectively.

3
3
For example, (11) = 11 × 11 × 11 = 1331, (14) = 14 × 14 × 14 = 2744,
(15) = 15 × 15 × 15 = 3375, (16) = 16 × 16 × 16 = 4096,
3
3
3
(19) = 19 × 19 × 19 = 6859
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