Property 1: Cubes of all even numbers
are even.
Example: 8 3 = 8 x 8 x 8 = 512
8 and 512 both are even numbers.
Property 2: Cubes of all odd numbers are odd.
Example: 9 3 = 9 x 9 x 9 = 729
9 and 729 both are natural odd numbers.
Property 3: The sum of the cubes of first n natural numbers is equal to the square of their sum.
1 3 + 2 3 + 3 3
+ … + n 3 = (1 + 2 + 3 + … + n) 2
Example: 1 3 + 2 3 + 3 3 + 4 3 = (1 + 2 + 3 + 4) 3
1 + 8 + 27 + 64 = (10) 2
100 = 100
Property
4: The cubes of
all numbers that end in 2 have 8 as the units digit. The cubes of all numbers
that end in 3 have 7 as the units digit.
Number
|
Cube number
|
7
|
343
|
8
|
512
|
17
|
4913
|
18
|
5832
|
Property 5 : Cubes of the numbers ending in digit 1, 4, 5, 6 and 9 are the numbers ending in the same digit.
Number
|
Cube
|
1
|
1
|
4
|
64
|
5
|
125
|
6
|
216
|
9
|
729
|
11
|
1331
|
14
|
2744
|
16
|
4096
|
19
|
6859
|
By
using property 4 and 5 cube and their
cube roots .
13 = 1 ( same number)
23 = 8 ( 2+8 = 10 )
33 = 27 ( 3+7 =10 )
43 = 64 ( same number)
53 = 125 ( same number)
63 = 216 ( same number)
73 = 343 (7+3 = 10)
83 = 512 (8 + 2= 10)
93 = 729 ( same number)
103 = 1000( same number)
Some interesting patterns:
1.
adding consecutive odd numbers:
1 = 1 = 13
3 + 5
= 8 =
23
7 + 9 + 11 =27 = 33
13 + 15+17+19 = 64 =43
Cubes and their prime factors:
74088 =
2*2*2*3*3*3*7*7*7
=23 * 33 * 73
= 2 * 3 * 7
= 42
Find
cube root of the number:
Example 17576
a) group last three digits in one group. 17576
b) take last group numbers 576 , there take last
units place. That is 6 , you know if units of
cube is 6 , cube root number also ends with 6
c)take 17 , which is below the 33 and more than 23 . Take less cube
root number ,that is 2.
So cube root of 74088 is 26.
Example
804357
a)804 357 , 357 where 7 is the
units place ,so units place of cube root is 3.
b)804 which is subtract 93(729) = difference is 75, and 804 subtract
from 103 (1000) difference is 271 ,which is far from “0” .
so we are taking 9 .
finally cube root of 804357 =933 .
Prime factorization Of number
|
Prime factorization of the cube number
|
4
= 2*2
|
43 = 64 =2*2*2*2*2*2
= 23 *23
|
6
= 2*3
|
63 = 216 = 2*2*2*3*3*3
= 23 *33
|
15
= 3*5
|
153
= 3375=3*3*3*5*5*5
= 33*53
|
12
= 2*2*3
|
123
= 1728 = 2*2*2*2*2*2*3*3*3
= 23*23*33
|
