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For certain ordered pairs (a,b) of real numbers, the system of equations [\begin{aligned} ax+by&=1 \\ x^2 + y^2 &= 50 \end{aligned}

has at least one solution, and each solution is an ordered pair (x,y) of integers. How many such ordered pairs (a,b) are there?

Dec 16, 2018

$$\text{there are only so many pairs of integers that satisfy the second equation}\\ \text{They are }(\pm 1, \pm 7),(\pm 7,\pm 1),(\pm 5,\pm 5)\\ \text{each of these produces 4 ordered pairs }(a,b) \text{ thus there are 12 total}$$