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

Jun 7, 2015

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

Possibility 1: the first integer is odd:
$\textcolor{w h i t e}{\text{XXXX}}$call the first integer $x$ (the four integers will be $x , x + 1 , x + 2 , \mathmr{and} x + 3$)

The sum of the even integers is
$\textcolor{w h i t e}{\text{XXXX}}$$\left(x + 1\right) + \left(x + 3\right) = 60$
$\rightarrow$$\textcolor{w h i t e}{\text{XXXX}}$$2 x + 4 = 60$
$\rightarrow$$\textcolor{w h i t e}{\text{XXXX}}$$x = 28$
which contradicts the assumption that the first integer is odd.

Possibility 2: the first integer is even:
$\textcolor{w h i t e}{\text{XXXX}}$call the first integer $x$ (the four integers will be $x , x + 1 , x + 2 , \mathmr{and} x + 3$)

The sum of the even integers is
$\rightarrow$$\textcolor{w h i t e}{\text{XXXX}}$$\left(x\right) + \left(x + 2\right) = 60$
$\rightarrow$$\textcolor{w h i t e}{\text{XXXX}}$$2 x + 2 = 60$
$\rightarrow$$\textcolor{w h i t e}{\text{XXXX}}$$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. (???)

Jun 7, 2015

Any even number $A$ can be of the form $A = 2 \cdot B$ 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:

$\sum \left(1 \rightarrow 8\right) = 36$

$\sum \left(1 \rightarrow 3\right) = 6$

$\Rightarrow \sum \left(4 \rightarrow 8\right) = 30$

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)