State whether the following statement is true or false.

The square of any positive integer cannot be of form 6b + 2 or 6b + 5 for any integer b.

Question

State whether the following statement is true or false.The square of any positive integer cannot be of form 6b + 2 or 6b + 5 for any integer b.

Accepted Answer

According to Euclid’s division lemma,Given positive integers say a and b, there exist unique integers q and r which satisfiesa=bq+r, where 0≤r<b. Let n be a positive integer. So, n=6q+r, and

0≤r<6
  Therefore, there exist six cases-If r=0, then n=6q

⇒
      n
      2
     
     =

6q
      
      2

⇒
      n
      2
     
     =36
      q
      2

⇒
      n
      2
     
     =6
      
       6
        q
        2

⇒
      n
      2
     
     =6b

Here,

b=6
    q
    2

is an integer and

n
    2

is divisible by 6.If r=1, then n=6q+1

⇒
      n
      2
     
     =

6q+1
      
      2

⇒
      n
      2
     
     =36
      q
      2
     
     +12q+1

⇒
      n
      2
     
     =6
      
       6
        q
        2
       
       +2q
     +1

⇒
      n
      2
     
     =6b+1

Here,

b=6
    q
    2
   
   +2q
   is an integer.This means, and

n
    2

is divisible by 6.If r=2, then n=6q+2.

⇒
      n
      2
     
     =

6q+2
      
      2

⇒
      n
      2
     
     =36
      q
      2
     
     +24q+4

⇒
      n
      2
     
     =6
      
       6
        q
        2
       
       +4q
     +4

⇒
      n
      2
     
     =6b+4

Here,

b=6
    q
    2
   
   +4q
  is an integer.This means, and

n
    2

is divisible by 6.If r=3, then n=6q+3.

⇒
      n
      2
     
     =

6q+3
      
      2

⇒
      n
      2
     
     =36
      q
      2
     
     +36q+9

⇒
      n
      2
     
     =6
      
       6
        q
        2
       
       +6q+1
     +3

⇒
      n
      2
     
     =6b+3

Here,

b=6
    q
    2
   
   +6q+1
   is an integer.This means, and

n
    2

is divisible by 6.If r=4, then n=6q+4.

⇒
      n
      2
     
     =

6q+4
      
      2

⇒
      n
      2
     
     =36
      q
      2
     
     +48q+16

⇒
      n
      2
     
     =6
      
       6
        q
        2
       
       +8q+2
     +4

⇒
      n
      2
     
     =6b+4

Here,

b=6
    q
    2
   
   +8q+2
   is an integer.This means, and

n
    2

is divisible by 6.If r=5, then n=6q+5.

⇒
      n
      2
     
     =

6q+5
      
      2

⇒
      n
      2
     
     =36
      q
      2
     
     +60q+25

⇒
      n
      2
     
     =6
      
       6
        q
        2
       
       +10q+4
     +1

⇒
      n
      2
     
     =6b+1

Here,

b=6
    q
    2
   
   +10q+4
   is an integer. This means, and

n
    2

is divisible by 6.Thus, it is proved that the square of any positive integer is not of form 6q+2 or 6q+5.Therefore, the given statement is true.Hence, option (1) is correct.

State whether the following statement is true or false.

The square of any positive integer cannot be of form 6b + 2 or 6b + 5 for any integer b.

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