Twenty-ThreePro Problems > Math > Number and Quantity > Number Theory > Diophantine Equations
x and y are positive integers such that xy + x = 23, Find all possible ordered pairs (x,y).
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The square of the difference of two numbers is subtracted from the square of the sum of two numbers. The result is 144.
If both of the numbers are positive integers, find all possible values for the sum of the two numbers.
Find all positive integers M and N such that M2 + MN = 7.
I have some strings of red lights, and some strings of green lights for my Christmas tree. All my strings of red lights have the same number of lights on them, and all my strings of green lights have the same number of lights on them.
All the strings of lights have at least 10 lights, and no more than 50 lights.
If I put all of the red-light strings and all the green-light strings on the tree, I would have a total of 278 lights. If I used all of my red-light strings, and only four of my green-light strings, I would have 159 lights.
How many strings of each do I have, and how many lights are on each string?
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