Question

please solve this question friends
Q7 and Q8

Answer 1

7. Let x be any positive integer and b = 3

Euclid's Lemma/ Algorithm

x = 3q + r, where q ≥ 0 and 0 ≤ r < 3.  Therefore, every number can be

represented as these three forms. There are three cases.


Case 1:

Let x=3q+0

x²=(3q)²

   =3(3q²)                                            Where m is an integer such that m = 3q²

x²=3m 


Case 2:

x=(3q+1)

x²=(3q+1)²

x²=9q²+6q+1
   
   =3(3q^2+2q)+1                           Where m is an integer such that m = 3q²+2q

x²=3m+1 


Case 3:

x=3q+2

x²=(3q+2)²

   =9q²+12q+4
   
   =9q²+12q+3+1

   =3(3q²+4q+1)+1                       Where m is an integer such that m = 3q²+4q+1

x²=3m+1


The square of any positive integer is of the form 3m or 3m+1 

____________________________________________________________


8. Let a be any positive integer and b = 3

Euclid's Lemma/ Algorithm

a = 3q + r, where q ≥ 0 and 0 ≤ r < 3  Therefore, every number can be

represented as these three forms. There are three cases.


Case 1: When a = (3q)³

                           = 9q³

                            = 3(3q³)                       Where m is an integer such that m = 3q³

Case 2: When a = 3q + 1,

a³ = (3q +1)³

a³ = 27q³ + 27q² + 9q + 1 

a³ = 9(3q³ + 3q² + q) + 1

a³ = 9m + 1                          Where m is an integer such that m = (3q³ + 3q² + q)


 Case 3: When a = 3q + 2,

a³ = (3q +2)³ 

a³ = 27q³ + 54q² + 36q + 8 

a³ = 9(3q³ + 6q² + 4q) + 8

a³ = 9m + 8                         Where m is an integer such that m = (3q³ + 6q² + 4q) 


Therefore, the cube of any positive integer is of the form 9m, 9m + 1, or 9m + 8.

____________________________________________________________


☺ ☺ ☺ Hope this Helps ☺ ☺ ☺