Problem Analysis #1

To solve the problem, we need to find the smallest natural numbers $n$ and $m$ such that $5^n > 10^9$ and $5^m + 4^m > 10^9$, respectively, and then compute $n + m$.

Step 1: Find $n$ such that $5^n > 10^9$.

We start by testing powers of 5:

• $5^1 = 5$
• $5^2 = 25$
• $5^3 = 125$
• $5^4 = 625$
• $5^5 = 3125$
• $5^6 = 15625$
• $5^7 = 78125$
• $5^8 = 390625$
• $5^9 = 1953125$

Since $5^9 = 1953125$ is the first power of 5 that exceeds $10^9 = 1000000000$, we have $n = 9$.

Step 2: Find $m$ such that $5^m + 4^m > 10^9$.

We test powers of 5 and 4 starting from $m = 9$ (since $5^9 = 1953125$ is already greater than $10^9$, and we need to check if adding $4^m$ makes the sum exceed $10^9$):

• For $m = 9$:
  • $5^9 = 1953125$
  • $4^9 = 262144$
  • $5^9 + 4^9 = 1953125 + 262144 = 2211769$, which is greater than $10^9$.

Thus, $m = 9$.

Step 3: Compute $n + m$.

Since $n = 9$ and $m = 9$, we have: $n + m = 9 + 9 = 18$

The final answer is $\boxed{18}$.