Problem Analysis #1

To solve the problem, we are asked to find two values:

• $n$: the smallest natural number such that $5^n > 10^9$
• $m$: the smallest natural number such that $5^m + 4^m > 10^9$

Then compute $n + m$.

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Step 1: Find $n$

We want the smallest $n$ such that:

$$5^n > 10^9$$

Taking logarithms (base 10):

$$n \cdot \log_{10}(5) > 9 \Rightarrow n > \frac{9}{\log_{10}(5)}$$

We know that \log_{10}(5) $$\approx$$ 0.69897, so:

$$n > \frac{9}{0.69897} \approx 12.87$$

Thus, the smallest integer $n$ satisfying this is:

$$n = 13$$

Let’s verify by computing powers:

• $5^{12} = 244,140,625 < 10^9$
• $5^{13} = 1,220,703,125 > 10^9$

So, $n = 13$ is correct.

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Step 2: Find $m$

Now we want the smallest $m$ such that:

$$5^m + 4^m > 10^9$$

Since both $5^m$ and $4^m$ are increasing functions in $m$, the sum $5^m + 4^m$ is also strictly increasing. So we test successive values of $m$ until the sum exceeds $10^9$.

We already know from earlier:

• $5^{12} = 244,140,625$
• $4^{12} = 16,777,216$
• $5^{12} + 4^{12} = 260,917,841 < 10^9$

Then:

• $5^{13} = 1,220,703,125$
• $4^{13} = 67,108,864$
• $5^{13} + 4^{13} = 1,287,811,989 > 10^9$

Therefore, the smallest $m$ such that $5^m + 4^m > 10^9$ is:

$$m = 13$$

This makes sense because $5^m$ dominates $4^m$, and since $5^{13} > 10^9$, adding $4^{13}$ only increases the total further.

Also, for all $m < 13$, the sum $5^m + 4^m$ is still less than $10^9$. For example, at $m = 12$, the sum is about 260 million — well below $10^9$.

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Final Step: Compute $n + m$

$$n = 13,\quad m = 13 \Rightarrow n + m = 26$$

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✅ Final Answer:

$$\boxed{26}$$