Page 56 - ICSE Math 8

(b) 3,375
3 3,375
Writing 3,375 as a product of its prime factors, we get 3 1,125
3,375 = 3 × 3 × 3 × 5 × 5 × 5 3 375
5 125
You can see that no number is left ungrouped. 5 25
So, 3,375 is a perfect cube. 5 5
Now, taking out one number from each group, we get 3 and 5. 1
Multiply 3 and 5 to get the number whose cube is 3,375.
Thus, 3 × 5 = 15, i.e., 3,375 is the cube of 15.

Example 2: What is the smallest number by which 2,160 should be divided to 2 2,160
make it a perfect cube?
2 1,080
Solution: Writing 2,160 as a product of its prime factors, we get 2 540
2,160 = 2 × 2 × 2 × 2 × 3 × 3 × 3 × 5 2 270

Grouping them into groups of three, you 3 135
can see that 2 and 5 are left ungrouped and Try These 3 45
should be removed to make 2,160 a perfect 1. Write the cubes of the first 3 15
cube. 10 natural numbers. 5 5
So, the smallest number by which 2,160 2. How many perfect cubes are 1
should be divided to make it a perfect cube there from 1 to 1,000?
is 2 × 5 = 10.

Properties of Cubes of Natural Numbers

I. The cube of every even number is even.
II. The cube of every odd number is odd.
III. The sum of the cubes of the first n-natural numbers is equal to the square of their sum, i.e.,
3
2
3
3
3
1 + 2 + 3 + ... + n = (1 + 2 + 3 + ... + n) .
IV. (a) Cubes of the numbers ending with 1, 4, 5, 6, 9 also end with the same digits.
(b) Cubes of the numbers ending with 2 end with 8.
(c) Cubes of the numbers ending with 8 end with 2.
(d) Cubes of the numbers ending with 3 end with 7.
(e) Cubes of the numbers ending with 7 end with 3.
(f) Cubes of the numbers ending with 0 end with three zeros.
Table of cubes of numbers from 1 to 20

3
3
3
Number (x) Cube (x ) Number (x) Cube (x ) Number (x) Cube (x )
1 1 8 512 15 3,375
2 8 9 729 16 4,096
3 27 10 1,000 17 4,913
4 64 11 1,331 18 5,832

5 125 12 1,728 19 6,859
6 216 13 2,197 20 8,000
7 343 14 2,744

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