The list $11,20,31,51,82$ is an example of an increasing list of five positive integers in which the first and second integers add to the third, the second and third add to the fourth, and the third and fourth add to the fifth. How many such lists of five positive integers have 124 as the fifth integer?
(A) 10
(B) 7
(C) 9
(D) 6
(E) 8
If the first positive integer in the list is $a$ and the second is $b$, then the third integer is $a+b$, the fourth is $b+(a+b)$ or $a+2 b$, and the fifth is $(a+b)+(a+2 b)$ or $2 a+3 b$.

Thus, we are asked to find the number of pairs of positive integers $a$ and $b$, where $a$ is less than $b$ (since the list is increasing), and for which $2 a+3 b=124$.

What is the largest possible value for $b$ ?

If $b=42$, then $3 b=3 \times 42=126$ which is too large since $2 a+3 b=124$. (Note that a larger value of $b$ makes $3 b$ even larger.)

If $b=41$, then $3 b=3 \times 41=123$.

However in this case, we get that $2 a=124-123=1$, which is not possible since $a$ is a positive integer.

If $b=40$, then $3 b=3 \times 40=120$ and so $2 a=4$ or $a=2$.

Thus, the largest possible value for $b$ is 40 .

What is the smallest value for $b$ ?

If $b=26$, then $3 b=3 \times 26=78$ and so $2 a=124-78=46$ or $a=23$.

If $b=25$, then $3 b=3 \times 25=75$.

However in this case, we get that $2 a=124-75=49$, which is not possible since $a$ is a positive integer.

If $b=24$, then $3 b=3 \times 24=72$ and so $2 a=124-72=52$ or $a=26$.

However, if the first integer in the list is 26 , then the second integer can not equal 24 since the list is increasing.

Smaller values of $b$ will give larger values of $a$, and so the smallest possible value of $b$ is 26 .

From the values of $b$ attempted thus far, we notice that when $b$ is an odd integer, $3 b$ is also odd (since the product of two odd integers is odd), and $124-3 b$ is odd (since the difference between an even integer and an odd integer is odd).

So when $b$ is odd, $124-3 b$ is odd, and so $2 a$ is odd (since $2 a=124-3 b$ ).

However, $2 a$ is even for every choice of the integer $a$ and so $b$ cannot be odd.

Conversely, when $b$ is even, $124-3 b$ is even (as required), and so all even integer values of $b$ from 26 to 40 inclusive will satisfy the requirements.

These values of $b$ are $26,28,30,32,34,36,38,40$, and so there are 8 such lists of five integers that have 124 as the fifth integer.

Here are the 8 lists: $2,40,42,82,124 ; 5,38,43,81,124 ; 8,36,44,80,124 ; 11,34,45,79,124$; $14,32,46,78,124 ; 17,30,47,77,124 ; 20,28,48,76,124 ; 23,26,49,75,124$.

ANSWER: (E)