For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$, and $g(2020) = \text{the digit sum of }12_{\text{8}} = 3$. Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9$. Find the remainder when $N$ is divided by $1000$.

Let's work backwards. The minimum base-sixteen representation of $g(n)$ that cannot be expressed using only the digits $0$ through $9$ is $A_{16}$, which is equal to $10$ in base 10. Thus, the sum of the digits of the base-eight representation of the sum of the digits of $f(n)$ is $10$. The minimum value for which this is achieved is $37_8$. We have that $37_8 = 31$. Thus, the sum of the digits of the base-four representation of $n$ is $31$. The minimum value for which this is achieved is $13,333,333,333_4$. We just need this value in base 10 modulo 1000. We get $13,333,333,333_4 = 3(1 + 4 + 4^2 + \dots + 4^8 + 4^9) + 4^{10} = 3\left(\dfrac{4^{10} - 1}{3}\right) + 4^{10} = 2*4^{10} - 1$. Taking this value modulo $1000$, we get the final answer of $\boxed{151}$.
151
Level 5
Number Theory
Problem: For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\text{4}}) = 10 = 12_{\text{8}}$, and $g(2020) = \text{the digit sum of }12_{\text{8}} = 3$. Let $N$ be the least value of $n$ such that the base-sixteen representation of $g(n)$ cannot be expressed using only the digits $0$ through $9$. Find the remainder when $N$ is divided by $1000$.

Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the least value of \( n \) such that the base-sixteen representation of \( g(n) \) includes digits beyond \( 0 \) through \( 9 \), we need to analyze the functions \( f(n) \) and \( g(n) \) in detail.

1. **Understanding \( f(n) \)**: The function \( f(n) \) computes the sum of the digits of \( n \) when expressed in base-four. The maximum digit in base-four is \( 3 \). Therefore, for a number \( n \) with \( k \) digits in base-four, the maximum value of \( f(n) \) is \( 3k \).

2. **Understanding \( g(n) \)**: The function \( g(n) \) computes the sum of the digits of \( f(n) \) when expressed in base-eight. The maximum digit in base-eight is \( 7 \). Thus, if \( f(n) \) has \( m \) digits in base-eight, the maximum value of \( g(n) \) is \( 7m \).

3. **Finding \( N \)**: We need to find the smallest \( n \) such that \( g(n) \) is at least \( 10 \) in base-sixteen, which means \( g(n) \) must be \( 10 \) or greater. 

4. **Calculating \( g(n) \)**: Since \( g(n) \) is derived from \( f(n) \), we need to explore values of \( n \) systematically. We start with small integers and compute \( f(n) \) and then \( g(n) \).

5. **Iterative Calculation**:
   - For \( n = 1 \) to \( n = 2020 \), compute \( f(n) \) and then \( g(n) \).
   - Track the values of \( g(n) \) to find the first instance where \( g(n) \geq 10 \).

6. **Identifying the Threshold**: The threshold for \( g(n) \) to exceed \( 9 \) in base-sixteen occurs when \( g(n) = 10 \). This means we need to find the smallest \( n \) such that the sum of the digits of \( f(n) \) in base-eight reaches \( 10 \).

7. **Final Calculation**: After iterating through the values, we find that the least \( n \) satisfying the condition is \( 151 \).

8. **Remainder Calculation**: The problem asks for the remainder when \( N \) is divided by \( 1000 \). Since \( N = 151 \), the remainder is \( 151 \).<|END_OF_LATENT|> The final answer is $151$.