Question

The product of three consecutive positive integers is divisible by 6 is this statement true or
false? Justify your answer.​

Answer 1

Answer:

Yes the given statement is true. A product of three consecutive positive integers is divisible by 6.

Explanation:

Consecutive numbers refers to the number in sequence, that is the next number should be 1 greater than the previous number.

If the first positive integer = a

Then,

The second positive integer = ( a + 1 )

The third positive integer = ( a + 2 )

Now, if the product of these consecutive positive integer is divisible by 6, then their product must be a multiple of 6, that is a constant value times 6.

Now, any positive integer can be written in this form: 6m + r, where 0 < r < 6

The number could be any of these forms:

6m, 6m + 1, 6m + 2, 6m + 3, 6m + 4, 6m + 5

Let's assume the first positive integer is 6m

6m( 6m + 1 )( 6m + 2 )

It is clearly visible that in this case, it is a multiple of 6 as it is in the form 6k. So, the product is divisible by 6.

Let's assume the first positive integer is 6m + 1

(6m + 1)( 6m + 2)( 6m + 3 )

(6m + 1) 2(3m + 1)3(2m + 1 )

6( 6m + 1)(3m + 1)(2m + 1 )

Here, it is visible that the product is a multiple of 6 as it is in the form 6k.So, the product number is divisible by 6.

Let's assume the first positive integer is 6m + 2

( 6m + 2)( 6m + 3)( 6m + 3)

2( 3m + 1) 3( 2m + 1)( 6m + 3)

6 (3m + 1)( 2m + 1)( 6m + 3)

Here, it is visible that the product is a multiple of 6 as it is in the form 6k. So, the product number is divisible by 6.

Let's assume the first positive integer is 6m + 3

( 6m + 3)( 6m + 4)( 6m + 5)

3( 2m + 1) 2( 3m + 2)( 6m + 5)

6( 3m + 1)( 3m + 1)( 6m + 5)

Here, it is visible that the product is a multiple of 6 as it is in the form 6k. So, the product number is divisible by 6.

Let's assume the first positive integer is 6m + 4

(6m + 4)( 6m + 5)( 6(m + 1 ))

6(6m + 4)( 6m + 5)(m + 1 )

Here, it is visible that the product is a multiple of 6 as it is in the form 6k. So, the product number is divisible by 6.

Let's assume the first positive integer is 6m + 5

( 6m + 5)[6(m + 1)][6(m + 2) + 1]

( 6m + 5) 6(m + 1) [6(m + 2) + 1]

6( 6m + 5) (m + 1) [6(m + 2) + 1]

Here, it is visible that the product is a multiple of 6 as it is in the form 6k. So, the product number is divisible by 6.

Now, we've tried with every possible number in terms of 6m + r, So we can conclude the product of any three positive consecutive integer is always divisible by 6.

Answer 2

Answer:

It is the correct answer.

Step-by-step explanation:

Hope this attachment helps you.