Page 153 - Math_Genius_V1.0_C8_Flipbook

E:\Working\Focus_Learning\Math_Genius-8\Open_Files\07_Chapter_6\Chapter_6
\ 06-Jan-2025 Bharat Arora Proof-6 Reader’s Sign _______________________ Date __________

Steps Example: Is 243 is a perfect cube?
1. Find the prime factors of the given number. 3 243
3 81
3 27
3 9
3 3
1
2. Group the equal factors into triplets. 3 × 3 × 3 × 3 × 3 = 3 × (3 × 3)
3
3. If all the factors can be put in groups of three equal factors, the Here, we are left with two factors
given number is a perfect cube, otherwise it is not. 3 × 3, so 243 is not a perfect cube.

Example 2: Is 27000 a perfect cube? 2 27000
2 13500
Solution: By prime factorisation, 27000 = 2 × 2 × 2 × 3 × 3 × 3 × 5 × 5 × 5 2 6750
= (2 × 2 × 2) × (3 × 3 × 3) × (5 × 5 × 5) 3 3375
3 1125
All prime factors of 27000 can be grouped into triples of same factors and 3 375
no factors are left ungrouped. 5 125

So, 27000 is a perfect cube. 5 25
5 5
1

Example 3: Is 13824 a perfect cube? If yes, find the number whose perfect cube is this. 2 13824

Solution: By prime factorisation, 13824 = 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 × 3 × 3 × 3 2 6912
2 3456
3
3
3
3
= (2) × (2) × (2) × (3) = (2 × 2 × 2 × 3) = (24) 3 2 1728
3
The prime factors of 13824 can be grouped into triples of same factors and no factors 2 864
are left ungrouped. So, 13824 is a perfect cube of 24. 2 432
2 216
2 108
Knowledge Desk 2 54
3 27
The number which are both perfect square and Quick Check 3 9
perfect cube are known as Sqube number. Is 49000 a perfect 3 3
For example, 1, 64, 729, …. cube? 1

Enrichment

Once a famous mathematician Prof. G. H. Hardy visited the great Indian
mathematician S. Ramanujan in a taxi with the number 1729. Hardy described the
number as a dull number. Ramanujan pointed out that 1729 was indeed an interesting
number because it could be expressed as a sum of two cubes in two different ways
as shown below:
(a) 1729 = 1728 + 1 = 12 + 1 3 (b) 1729 = 1000 + 729 = 10 + 9 3
3
3
It is the smallest number which can be expressed as the sum of two different cubes
in two different ways. S.Ramanujan
1729 has since been known as the Hardy-Ramanujan number.
Explore through the internet or books and find at least two numbers which can be expressed as the
sum of cubes in two different ways.

151 Cubes and Cube Roots