Question

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

Answer 1

Step-by-step explanation:

Show that one and only one out of n, n+2, n+4 is divisible by 3 ... n+2 = 3q+2+2 =3q+4 = 3(q+1)+1 is not divisible by 3. n+4 = 3q+2+4 = 3q+6 = 3(q+2) is divisible by 3. thus one and only one out of n , n+2, n+4 is divisible by 3.

Answer 2

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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 (possible values of r = 0,1,2)

➝ n = 3q, or n = 3q + 1, or n = 3q + 2

$\underbrace{\bf\:Case\:I}$

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

$\underbrace{\bf\:Case\:II}$

If n = 3q + 1,then (n + 2) = 3q + 3 = 3(q + 1) which is divisible by 3.

Hence,In this case (n + 2) is divisible by 3.

$\underbrace{\bf\:Case\:III}$

If n = 3q + 2,then (n + 4) = 3q + 6 = 3(q + 2) which is divisible by 3.

Hence,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 where n is any positive integers.