# Find the sum of all values of positive integer a

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• anemone
In summary, the purpose of finding the sum of all values of positive integer a is to determine the total value when all positive integers from 1 to a are added together. The sum of all values of positive integer a can be found using the formula (a * (a+1)) / 2. It is not possible for the sum of all values of positive integer a to be a negative number, as it always involves adding positive integers together. Additionally, the sum of all values of positive integer a can only be found if a is a positive integer and cannot be calculated with fractions or decimals. Finally, if a is equal to 0, then the sum of all values of positive integer a will also be 0 as there are
anemone
Gold Member
MHB
POTW Director
For a pair of positive integers $(a,\,b)$, $Q(a,\,b)$ is defined by

$Q(a,\,b)=\dfrac{a^2b+2ab^2-5}{ab+1}$.

Let $(a_1,\,b_1),\,(a_2,\,b_2),\,\cdots, (a_n,\,b_n)$ be all pairs of positive integers such that $Q(a,\,b)$ is an integer. Calculate $\displaystyle \sum_{i=1}^n a_i$,

If $ab+1$ divides $a^2b + 2ab^2 - 5$ then it also divides $(a+2b)(ab+1) - (a^2b + 2ab^2 - 5) = a+2b+5$.

So suppose that $a+2b+5 = k(ab+1)$ for a positive integer $k$. Then $k^2ab - ka - 2kb = 5k - k^2$. Therefore $$(ka-2)(kb-1) = 2 + 5k - k^2.$$ If $k=1$ then $(a-2)(b-1) = 6$. The four possible factorisations of $6$ give solutions $(a,b) = (3,7),\, (4,4),\, (5,3),\, (8,2)$.

If $k=2$ then $(2a-2)(2b-1) = 8$, or $(a-1)(2b-1) = 4$, giving only one solution $(a,b) = (5,1)$ (because $2b-1$ must be odd).

If $k=3$ then $(3a-2)(3b-1) = 8$, giving solutions $(1,3)$ and $(2,1)$.

If $k=4$ then $(4a-2)(4b-1) = 6$, or $(2a-1)(4b-1) = 3$, giving the solution $(1,1)$.

If $k=5$ then $(5a-2)(5b-1) = 2$, which has no solutions in positive integers.

If $k\geqslant6$ then $2+5k-k^2$ is negative, so there can be no more solutions.

So in total there are eight pairs of positive integers for which $Q(a,b)$ is an integer, namely $$(a,b) = (1,1),\ (1,3),\ (2,1),\ (3,7),\ (4,4),\ (5,1),\ (5,3),\ (8,2).$$ The sum of their $a$-coordinates is $\displaystyle\sum_{i=1}^8 a_i = 1+1+2 +3 +4 +5 +5 +8 = 29.$

## 1. What is the problem statement?

The problem is asking to find the sum of all values of positive integer a.

## 2. What is the range of values for a?

The range of values for a is all positive integers.

## 3. What is the formula for finding the sum of all values of positive integer a?

The formula is (a * (a + 1)) / 2, where a is the highest value of positive integer in the range.

## 4. How do you solve this problem?

To solve this problem, you need to first determine the highest value of positive integer a in the range. Then, plug that value into the formula (a * (a + 1)) / 2 to find the sum.

## 5. Can this problem be solved using a programming language?

Yes, this problem can be solved using a programming language by creating a loop to iterate through all the values of positive integer a and using the formula to calculate the sum.

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