Question

show that one and only one of n, n+2,n+4 is divisible by 3​

Answer 1

Correct Question :-

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

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$\huge\bf\underline \purple{Solution:-}$

On dividing n by 3,Let q be the quotient and r be the remainder.

Then,

• n = 3q +r, where 0 ≤ r < 3

⠀⠀⠀⠀⠀➝ n = 3q + r ,

• where r = 0,1, 2

⠀⠀⠀⠀⠀➝ n = 3q

⠀⠀⠀⠀⠀➝ n = 3q + 1

⠀⠀⠀⠀⠀➝ n = 3q + 2

Case 1st :-

If n = 3q ,then n is divisible by 3.

Case 2nd :-

If n = 3q +1 then,

⠀⠀⠀⠀⠀➝(n + 2) = 3q+3

⠀⠀⠀⠀⠀➝ 3(q + 1)

which is divisible by 3 .

So, in this case ,(n +2) is divisible by 3.

Case 3rd :-

When n = 3q +2 Then,

⠀⠀⠀⠀⠀➝ (n+4) = 3q +2 +4

⠀⠀⠀⠀⠀➝ 3q +6

⠀⠀⠀⠀⠀➝ 3 (q+2)

which is divisible by 3.

So, in this case , (n+4) is divisible by 3.

Hence, one and only one out of n, n+2 ,n +4 is divisible by 3.