Question

For consecutive integers, the sum of the even integers is 60. What is the value of the smallest odd integer?

Answer 1

Assuming you meant "Four (`4`) consecutive integers...",

Possibility 1: the first integer is odd:
`color(white)("XXXX")`call the first integer `x` (the four integers will be `x, x+1, x+2, and x+3`)

The sum of the even integers is
`color(white)("XXXX")` `(x+1)+(x+3) = 60`
`rarr` `color(white)("XXXX")` `2x+4 = 60`
`rarr` `color(white)("XXXX")` `x = 28`
which contradicts the assumption that the first integer is odd.

Possibility 2: the first integer is even:
`color(white)("XXXX")`call the first integer `x` (the four integers will be `x, x+1, x+2, and x+3`)

The sum of the even integers is
`rarr` `color(white)("XXXX")` `(x)+(x+2) = 60`
`rarr` `color(white)("XXXX")` `2x+2=60`
`rarr` `color(white)("XXXX")` `x= 29`
which contradicts the assumption that the first integer is even.

There is no solution to the given conditions (if my interpretation is correct).

Perhaps you intended something else. (???)

Answer 2

Any even number `A` can be of the form `A= 2*B` with B an ordinary number.

Therefore the sequence of consecutive even integers with a sum of 60 can be found by solving the sequence of consecutive integers with a sum of 30:

`sum(1rarr8) = 36`

`sum(1rarr3) = 6`

`rArr sum(4rarr8) = 30`

So we find that `4+5+6+7+8=30`
therefore `8+10+12+14+16=60`

Now we know the even numbers of the sequence of consecutive integers. Therefore, the smallest odd integer of the sequence will be the one coming before the smallest even integer: `7`

The sequence is `7;8;9;10;11;12;13;14;15;16(;17)`