The sum of two positive numbers is 1068 and their Highest Common Factor is 89. How many pairs of such numbers are there?
Let the two integers be A and B.
It is given that hcf(or, gcd) of A and B is 89.
Accordingly, there exist positive integers c and d such that:
A=89*c, and B=89*d, where gcd(c,d)=1
Now, A+B =1068
=> 89 (c+d)= 1068
=> c+d=12
Hence (disregarding the order of c and d) the only solution to the above equation with gcd(c,d) =1 occurs whenever, (c,d) =(1,11), (5,7), giving:
(A,B) =(89, 979), (445, 623)
Consequently, disregarding order, there are precisely 2 pairs of positive integers that satisfy all the conditions of the given puzzle.
Puzzle Solution
