Question

the cube of any positive integer cannot be of the form
option
7m,7m+1,7m+3,7m+6​

Answer 1

the cube of any positive integer cannot be of the form

option

7m,7m+1,7m+3,7m+6

Answer 2

Given : cube of any positive integer

To Find : cannot be in the form

1) 7m

2) 7m + 1

3) 7m + 3

4) 7m + 6

Solution:

 Any positive  integer can be represented in the form  a = bq +  r  

Without losing generality  Hence any number can be of the form :

7k , 7k + 1 , 7k + 2 , 7k + 3 , 7k + 4 , 7k + 5 , 7k + 6

Lets cube each number

(7k)³  = 7 * 49k³ = 7m    

(a + b)³  = a³ + b³  + 3ab(a + b)

(7k + 1)³ = (7k)³ + 1  + 3*7k *(7k + 1)  = 7k ( 49k² + 21k +3)  + 1  = 7m + 1

(7k + 2)³ = (7k)³ + 8  + 3*7k*2 *(7k +3)  = 7k ( 49k² + 42k + 18)  + 7 + 1

=  7k ( 49k² + 42k + 18 + 1 )  + 1

= 7m + 1

(7k + 3)³ = (7k)³ + 27  + 3*7k*3 *(7k +3)  = 7k ( 49k² + 63k + 27)  + 21 + 6

=  7k ( 49k² + 63k + 27 + 31 )  + 6

= 7m + 6

Similarly  (7k + 4)³ = 7m + 1

 (7k + 5)³ = 7m + 6

 (7k + 6)³ = 7m + 6

7m , 7m + 1, 7m + 6  are three forms of   cube of any positive integer

Hence cube of any positive integer cannot be in the form 7m + 3  from the given options

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