1. **Problem statement:** Find the number of ordered integer pairs $(x,y)$ satisfying the equation $$\frac{5}{x} + \frac{1}{y} = \frac{1}{18}.$$\n\n2. **Rewrite the equation:** Multiply both sides by $18xy$ (assuming $x,y\neq0$) to clear denominators:\n$$18xy \times \left(\frac{5}{x} + \frac{1}{y}\right) = 18xy \times \frac{1}{18}$$\nwhich simplifies to\n$$18y \times 5 + 18x \times 1 = xy$$\nor\n$$90y + 18x = xy.$$\n\n3. **Rearrange:** Bring all terms to one side:\n$$xy - 18x - 90y = 0.$$\n\n4. **Add and subtract $18 \times 90$: $$xy - 18x - 90y + 1620 = 1620.$$\n\n5. **Factor as $(x-90)(y-18)$:** Observe that\n$$(x-90)(y-18) = xy - 18x - 90y + 1620,$$\ntherefore we have\n$$(x-90)(y-18) = 1620.$$\n\n6. **Look for integer solutions:** For each integer divisor $d$ of 1620, we can set\n$$x - 90 = d \quad \text{and} \quad y - 18 = \frac{1620}{d}.$$\nHere, $d$ divides 1620 and $d \neq 0$. $x$ and $y$ are integers if and only if $d$ is an integer divisor of 1620.\n\n7. **Count divisors of 1620:** Factor 1620:\n$$1620 = 2^2 \times 3^4 \times 5^1.$$\nThe number of positive divisors is given by multiplying one plus each exponent:\n$$(2+1)(4+1)(1+1) = 3 \times 5 \times 2 = 30.$$\n\n8. **Total divisors:** Since divisors can be positive or negative, the total number of integer divisors is $2 \times 30 = 60$.\n\n9. **Check whether all correspond to valid $x,y$ pairs:** Since any divisor $d$ of 1620 gives unique $x,y$, but need to avoid division by zero in the original equation ($x \neq 0$, $y \neq 0$). Here, $x=90+d$, $y=18+\frac{1620}{d}$. Must check if any divisor $d$ causes $x=0$ or $y=0$.\n\n- If $x=0$, then $90 + d = 0 \Rightarrow d = -90$.\n- If $y=0$, then $18 + \frac{1620}{d} = 0 \Rightarrow \frac{1620}{d} = -18 \Rightarrow d = -90$.\nSo $d = -90$ causes $x=0$ and $y=0$ simultaneously which is not allowed.\n\n10. **Exclude $d = -90$:** One divisor to exclude from the 60 total divisors.\n\n11. **Final answer:** $60 - 1 = 59$ ordered pairs of integers satisfy the equation.