Question

Find the number of ordered pairs $(a,b)$ of integers such that
\[\frac{a + 2}{a + 5} = \frac{b}{4}.\]

Answer 1

Cross-multiply: 4(a+2) = b(a+5) ⇒ ab - 4a + 5b - 8 = 0 ⇒ (b-4)a = 8 - 5b.

So for b ≠ 4,
a = (8-5b)/(b-4) = -5 - 12/(b-4).

Thus a is integer iff b-4 divides 12. The (nonzero) divisors of 12 are ±1, ±2, ±3, ±4, ±6, ±12, giving 12 possible values of b (b ≠ 4). Each yields a corresponding integer a, and none gives a = −5 (which would make the original denominator zero).

Therefore there are 12 ordered pairs (a,b).