How many ordered quadruples of distinct positive integers (a,b,c,d) are there such that 1/a+1/b+1/c+1/d=1?
To determine the number of ordered quadruples (a, b, c, d) that satisfy the equation 1/a + 1/b + 1/c + 1/d = 1, we can follow the steps below:
Step 1: Initial Assumptions
We need to find distinct positive integers (a, b, c, d), so let's assume a < b < c < d. This assumption will help us avoid duplicate solutions.
Step 2: Simplify the Equation
To make the equation more manageable, we will start by multiplying both sides by abcd to eliminate the denominators:
bcd + acd + abd + abc = abcd
Step 3: Rearranging and Factoring
Rearrange the terms in the equation:
ab(cd) + ac(bd) + ad(bc) + bc(ad) = abcd
Now, factor out the common factors:
(cd)(ab + ac + ad + bc) = abcd
Step 4: Deducing Constraints
To find the number of ordered quadruples, we need to determine the constraints on the values of a, b, c, and d.
From the equation above, we can deduce that (ab + ac + ad + bc) must divide abcd, and since a < b < c < d, we can conclude that ab + ac + ad + bc > abcd.
Step 5: Analyzing the Factors
Let’s analyze the divisors of abcd by considering four cases:
Case 1: (ab + ac + ad + bc) = a × b × c × d
In this case, the numerator on the left side of the equation equals abcd, resulting in a solution.
Case 2: (ab + ac + ad + bc) = 2 × a × b × c × d
Here, the numerator is twice the denominator, meaning there are two possible solutions.
Case 3: (ab + ac + ad + bc) = 3 × a × b × c × d
Similar to Case 2, we have three possible solutions.
Case 4: (ab + ac + ad + bc) > 4 × a × b × c × d
In this case, the numerator is greater than four times the denominator, resulting in no solutions.
Step 6: Calculating the Number of Solutions
To determine the total number of ordered quadruples, we need to calculate the number of solutions for all the cases considered above.
Case 1: There is one solution for this case.
Case 2: There are two solutions in this case.
Case 3: There are three solutions for this case.
Finally, by adding the solutions from all cases, we can determine the total number of ordered quadruples that satisfy the equation.
Please note that it is difficult to further simplify the equation to obtain an exact formula for the number of solutions without making additional assumptions or using computational methods.
Therefore, the total number of ordered quadruples of distinct positive integers (a, b, c, d) that satisfy the equation 1/a + 1/b + 1/c + 1/d = 1 depends on the values of a, b, c, and d, as determined by the analysis of the divisors of abcd.