Problem Analysis #1

To find the value of $n+m$, we first determine $n$ and $m$ individually.

1. Find $n$: We want the smallest natural number $n$ such that: $5^n > 10^9$

We can rewrite $10^9$ as: $10^9 = (2 \times 5)^9 = 2^9 \times 5^9 = 512 \times 5^9$

So we need: $5^n > 512 \times 5^9 \implies 5^{n-9} > 512$

Let's evaluate powers of 5:

• For $n-9 = 3$: $5^3 = 125 \le 512$
• For $n-9 = 4$: $5^4 = 625 > 512$

Thus, the smallest integer $n-9$ is $4$, which gives: $n = 13$

2. Find $m$: We want the smallest natural number $m$ such that: $5^m + 4^m > 10^9$

Since $5^{13} > 10^9$, we know that $5^{13} + 4^{13} > 10^9$, so $m \le 13$.

Let's check if $m = 12$ satisfies the inequality:

• $5^{12} = 244,140,625$
• $4^{12} = 16,777,216$

Adding these values: $5^{12} + 4^{12} = 244,140,625 + 16,777,216 = 260,917,841$

Since $260,917,841 < 1,000,000,000$, $m = 12$ is too small.

Thus, the smallest natural number $m$ is: $m = 13$

3. Calculate $n+m$: $n + m = 13 + 13 = 26$

\boxed{26}