Question

show that one and only one out of n, n+2,or n+4 is divisible by 3 where n is a positive integer​

Answer 1

$\huge\boxed{\fcolorbox{blue}{orange}{HELLO\:MATE}}$

GIVEN:

★ Three numbers n, n +2, n+4 .

TO PROVE:

★Only one of them is divisible by 3 .

PROOF:

Let us take that n is divided by 3.

Then by Euclid division Lemma,

$\large\green{\boxed{ n = 3q + r}}$

For some positive integers q and r.

______________________________________

Clearly r< b

=>$r< 3$

$\large\red{\boxed{ r\epsilo-∞, 3) }}$

But r is non negative,

So, 0≤r<3

Therefore r =0, 1,2

______________________________________

★Case 1:

When r = 0

=>$n =3q$

(divisible by 3)

....................

=>$n+1=3q+1$

(not divisible by 3)

....................

$n+2=3q+2$

(not divisible by 3)

_____________________________________

★Case 2:

When r = 1.

=>$n =3q+1$

(not divisible by 3)

.........................

=>$n +1=3q+2$

(not divisible by 3)

.......................

=>$n+2 =3q+3$

=>$n+2=3(q+1)$

(divisible by 3)

______________________________________

★Case 3:

When r = 2

=>$n =3q+2$

(not divisible by 3)

.............................

=>$n +1=3q+3$

=>$n+1=3(q+1)$

(divisible by 3)

................................

=>$n +2=3q+4$

(not divisible by 3)

______________________________________

Hence in every cases only one was found to be divisible by 3.

Hence proved.

$\huge\orange{\boxed{HENCE \:PROVED}}$