4.41. Find all natural numbers $m$ for which: a) $3^{m}-1$ is divisible by $2^{m}$; b) $31^{m}-1$ is divisible by $2^{m}$.

## 4.8. Remainders of Division
4.41. a) Answer: 1, 2, and 4. Using the solution to problem 4.40 b). If \( a=3 \), then \( d(a-1)=1 \) and \( d\left(a^{2}-1\right)=3 \). The inequality \( s \leqslant \frac{2+n}{2^{n}} \) has the following solutions: \((s, n)=(1,1)\) and \((1,2)\).

b) Answer: 1, 2, 4, 6, and 8. If \( a=31 \), then \( d(a-1)=1 \) and \( d\left(a^{2}-1\right)=d(30 \cdot 32)=6 \). The inequality \( s \leqslant \frac{5+n}{2^{n}} \) has the following solutions: \((s, n)=(1,1),(3,1),(1,2)\) and \((1,3)\). (Recall that the number \( s \) is odd.)