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State true or false:
The product of any three consecutive natural numbers is divisible by 6.

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True

False

answer is A.

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Detailed Solution

We have to check whether the product of any three consecutive natural numbers is divisible by 6".
Let three consecutive natural numbers be

n, n + 1 and n + 2

.
If the number is divisible by 6 then it is divisible by 2 and 3.
Let's consider that the number is divisible by 3.
The remainder obtained is either 0 or 1 or 2 in case the number is divided by 3.
So, let

n = 3 a, 3 a + 1, 3 a + 2

where a is some integer.
If

n = 3 a

, n is divisible by 3.
If

n = 3 a + 1

. Then,

n + 2 = 3 a + 1 + 2

\Rightarrow n + 2 = 3 a + 3

\Rightarrow n + 2 = 3 (3 a + 1)

\Rightarrow n + 2

is divisible by 3.
If

n = 3 a + 2

. Then,

n + 1 = 3 a + 2 + 1

\Rightarrow n + 1 = 3 a + 3

\Rightarrow n + 1 = 3 (a + 1)

\Rightarrow n + 1

is divisible by 3.
So,

n (n + 1) (n + 2)

is divisible by 3.
Let's consider that the number is divisible by 2.
The remainder obtained is either 0 or 1 in case the number is divided by 2.
So, let

n = 2 b, 2 b + 1

where a is some integer.
If

n = 2 b

, then n and

n + 2

is divisible by 2.
If

n = 2 b + 1

. Then,

\Rightarrow n + 1 = 2 b 1 + 1

\Rightarrow n + 1 = 2 b + 2

\Rightarrow n + 1 = 2 (b + 1)

n + 1

is divisible by 2.
So, C is divisible by 2.
Since,

n + 2 = 3 a + 1 + 2

is divisible by 6 then it is divisible by 2 and 3.
Therefore, it is true that the product of any three consecutive natural numbers is divisible by 6.
Hence, option 1 is correct.

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