Let a and b be two positive integers, where a ≥ b. Find all pairs a, b such that their sum, their positive difference, their product, and their quotient add to 36
a+b + a-b + ab + a/b = 36
a(2 + b + 1/b) = 36
So, a is a factor of 36
Also, note that b + 2 + 1/b = (√b + 1/√b)^2
Looks like (2 + b + 1/b) must be a perfect square, so 1,4,9
I guess the only pair I see is a=9, b=1
Thanks
interpretation:
(a+b) + (a-b) + (ab) + (a/b) = 36
2a + ab + a/b = 36
2ab + ab^2 + a = 36b
ab^2 + a = 36b - 2ab
a(b^2 + 1) = 2b(18 - a)
a/(18-a) = 2b/(b^2 + 1)
I tried b = 1, then a = 9
b = 2, then a = 8
b = 5, then a = 5
could not find any others.
so (9,1), (8,2), (5,5)
testing: e.g. (8,2)
8+2 + 8-2 + (8)(2) + 8/2
= 16 + 16 + 4
= 36
Nice save
To find all pairs (a, b) such that their sum, positive difference, product, and quotient add up to 36, we can start by writing the mathematical expressions for each of these operations and setting up equations.
Let's go through each operation one by one:
1. Sum: The sum of a and b is a + b.
2. Positive difference: The positive difference between a and b is a - b. Since it is specified that a ≥ b, the positive difference is simply a - b.
3. Product: The product of a and b is a * b.
4. Quotient: The quotient of a and b is a / b. However, note that the quotient may not always be an integer since it is specified that a and b are positive integers.
Now, let's set up the equations using the expressions above:
a + b + (a - b) + (a * b) + (a / b) = 36
Simplifying the equation further, we get:
a + b + a - b + a * b + a / b = 36
2a + a * b + a / b = 36
To find all possible pairs (a, b), we need to systematically check different values of a and b to see if they satisfy the equation. Since a and b are positive integers, start by selecting suitable values for a and b, then substitute them into the equation to see if the sum, positive difference, product, and quotient satisfy the equation. Iterate through different values until all possibilities have been exhausted.
For example, you can start by trying a = 3 and b = 1:
2*3 + 3*1 + 3/1 = 9 + 3 + 3 = 15, which is less than 36.
You can continue this process by trying different values of a and b to find all pairs that satisfy the equation.
Note: This is a manual and exhaustive method to find all pairs (a, b). If you want to use a more efficient approach and automate the process, you can write a computer program or use a mathematical software/tool to solve the equation and find all the pairs (a, b).