State whether the following statement is true or false.

One of any three consecutive positive integers must be divisible by 3.

Question

State whether the following statement is true or false.One of any three consecutive positive integers must be divisible by 3.

Accepted Answer

According to Euclid’s division lemma, we know,Given positive integers say a and b, there exist unique integers q and r which satisfies a=bq+r, where 0≤r<b. Let q be the quotient and r be the remainder when n is divided by 3.Following that, it is expressed as, using Euclid's division algorithm.n=3q+r, where

0≤r<3
  Therefore, there exist three cases, If r=0, then n=3q. Which means n is divisible by 3. Adding 1 on both sides,

⇒n+1=3q+1
  This means n+1 is not divisible by 3.After adding 2 on both sides, we get,

⇒n+2=3q+2
  This means n+2 is not divisible by 3.Consequently, only n, in this case, is divisible by 3.For r=1n=3q+1which is not divisible by 3.Adding 1 on both sides,

⇒n+1=3q+1+1

⇒n+1=3q+2

This means n+1 is not divisible by 3. Now, we will add 2 on both sides.

⇒n=3q+1

⇒n+2=3q+1+2

⇒n+2=3q+3

⇒n+2=3
      
       q+1

Hence, n+2 is divisible by 3.Hence, in this case, only n+2 is divisible by 3. For r=2, n=3q+2, which is not divisible by 3.Adding 1 on both sides,

⇒n+1=3q+2+1

⇒n+1=3q+3

⇒n+1=3
      
       q+1

Hence, n+1 is divisible by 3. Now, we will add 2 on both sides.

⇒n+2=3q+2+2

⇒n+2=3q+4

⇒n+2=3q+3+1

⇒n+2=3
      
       q+1
     +1

Hence, n+2 is not divisible by 3. Hence, in this case, only n+1 is divisible by 3. Thus, it is proved that one of

n,n+1
   and

n+2
   is divisible by 3.So, it is shown that any three consecutive numbers can be divided by three.Therefore, the given statement is true.Hence, option (1) is correct.

State whether the following statement is true or false.

One of any three consecutive positive integers must be divisible by 3.

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