State true or false.The cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

Question

Accepted Answer

Given a statement the cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8. Assuming a as the positive integer whose cube is to be calculated.And let b be also a positive integer.According to Euclid’s division lemma, the relation of a and b is

a=bq+r
  .    Here, r = remainder and q = quotient. Assuming the value of b = 3. Possible remainders are 0,1,2.Putting

r=0 
  in the above equation,

a=3×q+0

a=3×q
  Now, taking the cube on both sides,

=
      (3q)
      3

=27
      q
      3

=9×3
      q
      3

=9m                 [m=3
      q
      3
     
     ]

Hence, it is of 9m form.Putting r = 1 in equation,

a=3×q+1
  .Now, taking the cube on both sides,Hence, it is of 9m + 1 form.Putting r = 2 in the equation,

a=3×q+2
  .Now, taking the cube on both sides,

=
    
     (3q+2)
    3

=
    
     (3q)
    3
   
   +3×

3q
     
    2
   
   ×2+3×3q×
    2
    2
   
   +
    2
    3
   
                 [As,

a+b
     
    3
   
   =
    a
    3
   
   +3
    a
    2
   
   b+3a
    b
    2
   
   +
    b
    3
   
   ]

=27
      q
      3
     
     +108
      q
      2
     
     +36q+8

=
      
       27
        q
        3
       
       +108
        q
        2
       
       +36q
     +8

=9×
      
       3
        q
        3
       
       +12
        q
        2
       
       +4q
     +8

=9m+8                                            [m=3
      q
      3
     
     +12
      q
      2
     
     +4q]

Hence, it is of 9m + 8 form.Therefore, proving the given statement is true.Hence, option 1 is correct.

State true or false.
The cube of any positive integer is of the form 9m, 9m + 1 or 9m + 8.

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