Question

. Use Euclid’s division lemma to show that the square of any positive integer is either of the form 3p or 3p+ 1, where p is an integer.

Answer 1

 



deepak asked in Math 

... division lemma to show that the cube of any positive integer ...

let a be any postivi integer which when divided by 3 gives q as quotient and r as remainder

according to euclids division lemme,a=3q+r where r=0,1 or 2

possible values of a: when r=0

then a=3q

  a3=27q3  CUBING BOTH SIDES

  a3=9m  where m = 3q3

when r=1 then a = 3q+1

  a3 = 27q3+1+27q2+9q  CUBING BOTH SIDES 

 a3 = 9m+1  where m = 3q3+3q2+q

when r = 2 

then a = 3q+2

 a3 = 27q3+8+54q2+18q 

  a3 = 9m+8  where m=3q3+6q2+2q

from all three cases we get cube of any postive integer is of form 9m,9m+1 or 9m+8

Answer 2

first take 3a 
=>(3a)$^{2}$ 
=> 9a$^{2}$

=> 3p  ( p = 3a$^{2}$ )

now take 3a + 1
=> (3a + 1)
=> 9a$^{2}$ + 6a + 1 

=> 3p + 1  ( p = 3a$^{2}$ + 2a )