Do you get 350?
Number Theory
Level
3
The sum of two positive integers is 350. Their GCD is 25.
How many unordered pairs of such positive integers are there?
Note : The order of the pair does not count. (x,y) is considered the same pair as (y,x)
The answer is 3.
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1 solution
Let the numbers be 25x and 25y where x and y are co-prime and y > x
By The Problem, 25x + 25y = 350 or 25(x+y)=350 or x+y = 14
Now, search for values of x and y which meet the following conditions:
There are 3 possible values of (x,y): (1,13), (3, 11), (5, 9)
Thus there are 3 such pairs: (25, 325), (75, 275) and (125, 225)