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
Starting with the smallest values of a,b,c,d we can see that
1/2 + 1/3 + 1/6 = 1
but that's only 3 fractions. So, if we can split 1/6 into two fractions, we are ok:
1/6
= 1/7 + 1/42
= 1/8 + 1/24
= 1/9 + 1/18
= 1/10 + 1/15
= 1/11 + 5/66 but that's not allowed
= 1/12 + 1/12 and we are done, since there are no other possibilities. That is, any higher divisors won't be unit fractions (except those already listed above)
That gives us 4 solutions starting with 1/2 + 1/3 + * + *
Now, try 1/2+1/4+*+* where *+*=1/4 and so on till you run out of possibilities.
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 by understanding the constraints on the values of a, b, c, and d. Since a ≤ b ≤ c ≤ d, we can assume that a = 1 to begin with. If a > 1, the left side of the equation would be greater than 1, which contradicts our equation.
So, we have 1/1 + 1/b + 1/c + 1/d = 1, which simplifies to:
1 + 1/b + 1/c + 1/d = 1.
Next, we can simplify the equation by subtracting 1 from both sides:
1/b + 1/c + 1/d = 0.
Now, let's analyze the behavior of the equation when b = 1. Substituting b = 1, we get:
1/1 + 1/c + 1/d = 0,
which simplifies to:
1 + 1/c + 1/d = 0.
Since c and d are positive integers, there is no possible value for c and d that would satisfy this condition. Therefore, there are no solutions when b = 1.
Moving on, let's consider the case when b = 2. Substituting b = 2, we have:
1/2 + 1/c + 1/d = 0.
By rearranging the equation, we get:
1/c + 1/d = -1/2.
To further simplify the equation, we can multiply both sides by 2cd:
2d + 2c = -cd.
Since c and d are positive integers, the left side of the equation will always be positive. However, the right side, -cd, will always be negative. Therefore, there are no solutions when b = 2.
We can continue this process for higher values of b. However, every time we increase the value of b, the left side of the equation will increase, while the right side will remain negative. Thus, we can conclude that there are no solutions for the given equation.
In summary, there are no solutions to the equation 1/a + 1/b + 1/c + 1/d = 1, where a, b, c, and d are positive integers and a ≤ b ≤ c ≤ d.