Question

The product of three consecutive integers is divisible by:
a) 4
b) 6
c) no common factor
d) only 1

[GIVE REASON]​

Answer 1

4

Step-by-step explanation:

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Answer 2

Answer:

6

Step-by-step explanation:

In the given questions, we have to prove that

the product of any three consecutive numbers is divisible by 6

. If a number is divisible by 6

, then it means that it is also divisible by 2

and 3

.

So, let us prove that the product of any three consecutive numbers is divisible by 2

and 3

.

Consider the three consecutive numbers to be x,(x+1),(x+2)

.

For 2

:

If x

is not divisible by 2

, then it means that x

is odd.

Hence, if x

is odd, then it is a known fact that x+1

(any odd number plus 1

) is even, hence, is divisible by 2

.

Thus, out of the three consecutive numbers, at least one of them is always divisible by 2

.

For 3

:

Now, if a number is not divisible by 3

and is divided by it, then it can leave either of only two remainder – 1

and 2

.

If x

is not divisible by 3

and when divided by 3

leaves a remainder of 1

, then x+2

is going to be divisible by 3

.

If x

is not divisible by 3

and when divided by 3

leaves a remainder of 2

, then x+1

is going to be divisible by 3

.

Similarly, for the other two cases – (x+1)

and (x+2)

not being divisible by 3

, we can have that either of the other two are going to be divisible by 3

.

Thus, out of the three consecutive numbers, exactly one of them is always divisible by 3

.

Now, we have shown that out of the three consecutive numbers, one of them is always divisible by 2

and 3

. Hence, we showed that the product of any three consecutive numbers is always divisible by 6

.

Now, let us show that using some examples:

Let the triplet be 13,14,15

.

Product of the three numbers is 13×14×15=2730

.

Now, 2730÷6=455

Hence, the product of the given triplet is divisible by 6

.

Let the triplet be 16,17,18

.

Product of the three numbers is 16×17×18=4896

.

Now, 4896÷6=816

Hence, the product of the given triplet is divisible by 6

.