Name: Applications of Euclid's Division Lemma
Uploaded: 2019-07-19
Duration: 18 min 17 s
Description: Can you prove that only one out of three consecutive positive integers is divisible by 3? Watch how, along with more examples of the applications of Euclid's...

Question

Prove that the product of three consecutive positive integers is divisible by

6 .

Accepted Answer

According to Euclid's division lemma, we have, any number $a$ is in the form of
$b q + r$ where $0 \leq r < b .$

e.g $a = b q + r;$ where, $0 \leq r < b .$

Let the three consecutive positive integers be $n, n + 1$ and $n + 2$ .

Whenever a number is divided by $3,$ the remainder obtained is either $0, 1$ or $2 .$

Therefore, $n = 3 q$ or $n = 3 q + 1$ or $3 q + 2,$ where $q$ is some integer.

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

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

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

So, we can say that one of the numbers among $n, n + 1$ and $n + 2$ is always divisible by $3$ that is: $n (n + 1) (n + 2)$ is divisible by $3 .$

Similarly, whenever a number is divided by $2$ , the remainder obtained is either $0$ or $1$ .

Therefore, $n = 2 p$ or $2 p + 1$ , where $p$ is some integer.

If $n = 2 p$ then $n$ and $n + 2 = 2 p + 2 = 2 (p + 1)$ is divisible by $2$ .

If $n = 2 p + 1,$ then $n + 1 = 2 p + 1 + 1 = 2 p + 2 = 2 (p + 1)$ is divisible by $2$ .

So, we can say that one of the numbers among $n, n + 1$ and $n + 2$ is always divisible by $2 .$

Since, $n (n + 1) (n + 2)$ is divisible by $2$ and $3$ .

Thus, $n (n + 1) (n + 2)$ is divisible by $6 .$

Hence proved.

Prove that the product of three consecutive positive integers is divisible by $6 .$

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Prove that the product of three consecutive positive integers is divisible by 6.

Important Questions on Real Numbers

Learn and Prepare for any exam you want !

Prove that the product of three consecutive positive integers is divisible by $6 .$