Two radioactive materials A and B have decay constants $$25\lambda$$ and $$16\lambda$$ respectively. If initially they have the same number of nuclei, then the ratio of the number of nuclei of B to that of A will be 'e' after a time $$\frac{1}{a\lambda}$$. The value of a is _____

We have two radioactive materials A and B with decay constants $$\lambda_A = 25\lambda$$ and $$\lambda_B = 16\lambda$$ respectively. Initially, both have the same number of nuclei $$N_0$$.

After time $$t$$, the number of nuclei remaining are: $$N_A = N_0 e^{-25\lambda t}, \quad N_B = N_0 e^{-16\lambda t}$$

The ratio is: $$\frac{N_B}{N_A} = \frac{e^{-16\lambda t}}{e^{-25\lambda t}} = e^{(25-16)\lambda t} = e^{9\lambda t}$$

We are told this ratio equals $$e$$ (Euler's number) at time $$t = \frac{1}{a\lambda}$$. So: $$e^{9\lambda \cdot \frac{1}{a\lambda}} = e$$

$$e^{9/a} = e^1$$

Comparing exponents: $$\frac{9}{a} = 1$$, which gives $$a = 9$$.

Hence, the correct answer is 9.