The probability that Janice does not buy an item online in any week is given by:

\(1 - 0.35 = 0.65\)

The probability that Janice does not buy any item online in n weeks is:

\((0.65)^n\)

The probability that Janice buys at least one item online in n weeks is greater than 0.99, so:

\(1 - (0.65)^n > 0.99\)

Rearranging gives:

\((0.65)^n < 0.01\)

Taking logarithms on both sides:

\(n \log(0.65) < \log(0.01)\)

Solving for n gives:

\(n > \frac{\log(0.01)}{\log(0.65)} \approx 10.69\)

Since n must be an integer, the smallest possible value of n is 11.