1. The problem is to verify if $\frac{13}{52} \times \frac{12}{52} = \frac{1}{17}$.\n\n2. Start by multiplying the numerators and denominators: $$\frac{13 \times 12}{52 \times 52} = \frac{156}{2704}.$$\n\n3. Simplify the fraction $\frac{156}{2704}$. Find the greatest common divisor (GCD) of 156 and 2704.\n\n4. The GCD of 156 and 2704 is 52. Divide numerator and denominator by 52: $$\frac{\cancel{156}^{3}}{\cancel{2704}^{52}} = \frac{3}{52}.$$\n\n5. The simplified product is $\frac{3}{52}$, which is not equal to $\frac{1}{17}$.\n\n6. Therefore, $\frac{13}{52} \times \frac{12}{52} \neq \frac{1}{17}$. The correct product is $\frac{3}{52}$.