Name: Fundamental Theorem of Arithmetic
Uploaded: 2019-07-19
Duration: 5 min 4 s
Description: A number can be represented in the form of the Product of their Primes. This method is also called the Fundamental Theorem of Arithmetic. Watch the video to...

Question

Prove that one and only one out of

n, (n + 2)

and

(n + 4)

is divisible by

3

, where

n

is any positive integer.

Accepted Answer

Let $a = n, b = n + 2$ and $c = n + 4$

$∴$ Order triplet is $(a, b, c)$ $= (n, n + 2, n + 4)$

Where, n is any positive integer i.e, $n = 1, 2, 3, . . .$

At $n = 1; (a, b, c) = (1, 1 + 2, 1 + 4) = (1, 3, 5)$

At $n = 2; (a, b, c) = (2, 2 + 2, 2 + 4) = (2, 4, 6)$

At $n = 3; (a, b, c) = (3, 3 + 2, 3 + 4) = (3, 5, 7)$

At $n = 4; (a, b, c) = (4, 4 + 2, 4 + 4) = (4, 6, 8)$

At $n = 5; (a, b, c) = (5, 5 + 2, 5 + 4) = (5, 7, 9)$

At $n = 6; (a, b, c) = (6, 6 + 2, 6 + 4) = (6, 8, 10)$

At $n = 7; (a, b, c) = (7, 7 + 2, 7 + 4) = (7, 9, 11)$

At $n = 8; (a, b, c) = (8, 8 + 2, 8 + 4) = (8, 10, 12)$

We observe that each triplet consist of one and only one number which is multiple of $3$ i.e., divisible by $3$ .
Hence, one and only one out of $n, (n + 2)$ and $(n + 4)$ is divisible by $3$ , where, $n$ is any positive integer.

Alternate Method

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

Then, $n = 3 q + r$ , where, $0 \leq r < 3$

$\Rightarrow n = 3 q + r,$ where $r = 0, 1, 2$

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

Case $I$ If $n = 3 q$ , then $n$ is only divisible by $3$ .

but $n + 2$ and $n + 4$ are not divisible by $3$ .

Case $I I$ If $n = 3 q + 1,$ then $(n + 2) = 3 q + 3 = 3 (q + 1)$ , which is only divisible by 3, but $n$ and $n + 4$ are not divisible by $3$ .

Case $I I I$ When $n = 3 q + 2,$ then $(n + 4) = 3 q + 6 = 3 (q + 2)$ , which is only divisible by $3$ , but $n$ and $(n + 2)$ are not divisible by $3$ . So, in this case, $(n + 4)$ is divisible by $3$ .

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

Prove that one and only one out of $n, (n + 2)$ and $(n + 4)$ is divisible by $3$ , where $n$ is any positive integer.

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

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

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Prove that one and only one out of $n, (n + 2)$ and $(n + 4)$ is divisible by $3$ , where $n$ is any positive integer.