Question

Prove the sum of three consecutive is divisible by three, that the sum of 5 consecutive integers is divisible by 5, but the sum of four consecutive integers is not divisible by 4

Answer 1

Proved

Explanation:

Solving (a):

Let the numbers be:
`x, x + 1, x + 2.`

Their sum is:

`Sum = x + x + 1 + x + 2`

Collect Like Terms

`Sum = x + x + x + 1 + 2`

`Sum = 3x + 3`

Divide sum by 3.

`Result = (Sum)/(3)`

`Result = (3x+3)/(3)`

`Result = (3(x+1))/(3)`

`Result = x + 1`

Hence, this is true because there is no fractional part after the division

Solving (b):

Let the numbers be:
`x, x + 1, x + 2,x+3,x+4`

Their sum is:

`Sum = x + x + 1 + x + 2+x + 3 + x + 4`

Collect Like Terms

`Sum = x + x + x + x + x+1 + 2 + 3 + 4`

`Sum = 5x+10`

Divide sum by 5.

`Result = (5x + 10)/(5)`

`Result = (5(x + 2))/(5)`

`Result = x + 2`

Hence, this is true because there is no fractional part after the division

Solving (c):

Let the numbers be:
`x, x + 1, x + 2,x+3`

Their sum is:

`Sum = x + x + 1 + x + 2+x + 3`

Collect Like Terms

`Sum = x + x + x + x + 1 + 2 + 3`

`Sum = 4x+6`

Divide sum by 4.

`Result = (4x + 6)/(4)`

Split

`Result = (4x)/(4) + (6)/(4)`

`Result = x + 1.5`

The 1.5 means that the sum can not be divisible by 4