1. Identify compound $X$

Given: $\mathrm{Na_2O + H_2O \to 2X}$

Sodium oxide reacts with water to form sodium hydroxide: $\mathrm{Na_2O + H_2O \to 2NaOH}$

So, $X = \mathrm{NaOH}$

Number of oxygen atoms in one formula unit of $X$: $\mathrm{NaOH} \text{ has } 1 \text{ oxygen atom}$

2. Identify compound $Y$

Given: $\mathrm{Cl_2O_7 + H_2O \to 2Y}$

$\mathrm{Cl_2O_7}$ is the anhydride of perchloric acid. On reaction with water: $\mathrm{Cl_2O_7 + H_2O \to 2HClO_4}$

So, $Y = \mathrm{HClO_4}$

Number of oxygen atoms in one formula unit of $Y$: $\mathrm{HClO_4} \text{ has } 4 \text{ oxygen atoms}$

3. Total number of oxygen atoms in $X$ and $Y$

$1 + 4 = 5$

Therefore, the required integer is: $\boxed{5}$

4. Comparison with stored correct answer

Stored correct answer = $5$

Our derived answer also = $5$, so they agree.