Find the number of solutions to the equation
1/a+1/b+1/c+1/d=1
where a, b, c, d are positive integers and a≤b≤c≤d.
To find the number of solutions to the equation 1/a + 1/b + 1/c + 1/d = 1, where a, b, c, d are positive integers and a≤b≤c≤d, we can use a systematic approach.
Let's start with some observations:
1. As a general rule, the larger the value of a fraction, the smaller its denominator. So, in order for the sum to be equal to 1, at most three of the four fractions can have a denominator greater than or equal to 4.
2. Since a ≤ b ≤ c ≤ d, the largest possible value for a is 4. This is because the minimum value for b, c, and d will be at least 4.
Now let's consider the cases for a:
Case 1: a = 1
If a = 1, the equation becomes 1/b + 1/c + 1/d = 3/4. We can simplify this equation to 1/b + 1/c + 1/d = 3/4 by multiplying through by 4bcd. Now, we need to find the number of solutions to this equation, where b, c, d are positive integers.
Case 2: a = 2
If a = 2, the equation becomes 1/b + 1/c + 1/d = 1/2. We can simplify this equation to 1/b + 1/c + 1/d = 1/2 by multiplying through by 2bcd. Now, we need to find the number of solutions to this equation, where b, c, d are positive integers.
Case 3: a = 3
If a = 3, the equation becomes 1/b + 1/c + 1/d = 1/3. We can simplify this equation to 1/b + 1/c + 1/d = 1/3 by multiplying through by 3bcd. Now, we need to find the number of solutions to this equation, where b, c, d are positive integers.
Case 4: a = 4
If a = 4, the equation becomes 1/b + 1/c + 1/d = 1/4. We can simplify this equation to 1/b + 1/c + 1/d = 1/4 by multiplying through by 4bcd. Now, we need to find the number of solutions to this equation, where b, c, d are positive integers.
For each case, we can use various methods to find the number of solutions. These methods may include generating functions, combinatorial techniques, or algebraic manipulation.
By finding the number of solutions for each case and summing them, we can determine the total number of solutions to the original equation.