Find the molecular formula of $N,N$-dimethylaminopentane.

Pentyl group attached to nitrogen: $C_5H_{11}$

Two methyl groups on nitrogen: $2 \times CH_3 = C_2H_6$

So total formula is: $C_7H_{17}N$

Find molar mass of $C_7H_{17}N$: $7\times 12 + 17\times 1 + 14 = 84 + 17 + 14 = 115\,\text{g mol}^{-1}$

Moles of compound in $57.5\,\text{g}$: $\frac{57.5}{115} = 0.5\,\text{mol}$

Combustion equation in Dumas method

In Dumas method, the organic compound is heated with excess $CuO$. Carbon forms $CO_2$, hydrogen forms $H_2O$, and nitrogen is liberated as $N_2$.

Let: $C_7H_{17}N + xCuO \rightarrow 7CO_2 + \frac{17}{2}H_2O + \frac{1}{2}N_2 + xCu$

Balance oxygen atoms on products side:

Oxygen needed for $7CO_2$: $7\times 2 = 14$

Oxygen needed for $\frac{17}{2}H_2O$: $\frac{17}{2}$

Total oxygen atoms needed: $14 + \frac{17}{2} = \frac{28+17}{2} = \frac{45}{2}$

Hence, $x = \frac{45}{2}$

So, $C_7H_{17}N + \frac{45}{2}CuO \rightarrow 7CO_2 + \frac{17}{2}H_2O + \frac{1}{2}N_2 + \frac{45}{2}Cu$

Moles of $CuO$ required for $0.5$ mol compound: $0.5 \times \frac{45}{2} = \frac{45}{4} = 11.25\,\text{mol}$

The question expresses answer as: $\underline{\hspace{1cm}} \times 10^{-2}$

Write $11.25$ mol in this form: $11.25 = 1125 \times 10^{-2}$

Therefore, the required integer is: $\boxed{1125}$

Comparison with stored answer

Derived answer = $1125$

Stored correct answer = $1125$

Hence, they agree.