Question

Q.5. Show that one and only one out of n, n+2 and n+4 is divisible by 3, where n is any
positive integer.

Answer 1

Answer:

By Euclid's division lemma, a=bq+r, 0≤r<b.

Let,

a=n

b=3

r=0,1,2

Case 1 (r=0):-

a=bq+r

n=3q   {divisible by 3}

n+2=3q+2   {not divisible by 3}

n+4=3q+4    {not divisible by 3}

Only n is divisible by 3 when r is equal to 0.

Case 2 (r=1):-

n=3q+1  {not divisible by 3}

n+2=3q+1+2

     =3q+3  {divisible by 3}

n+4=3q+1+4

     =3q+5   {not divisible by 3}

Only n+2 is divisible by 3 when r is equal to 1.

Case 3 (r=2):-

n=3q+2   {not divisible by 3}

n+2=3q+2+2

     =3q+4   {not divisible by 3}

n+4=3q+2+4

     =3q+6   {divisible by 3}

Only n+4 is divisible by 3 when r is equal to 2.

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