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MATH / For every positive integer n, let mod₅(n) be the remainder obtained when n is divided by 5. Define a function f:{0,1,2,3,…}×{0,1,2,3,4}→{0,1,2,3,4} recursively as follows: …

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For every positive integer nn, let mod5(n)\text{mod}_5 (n) be the remainder obtained when nn is divided by 5. Define a function f:{0,1,2,3,…}×{0,1,2,3,4}→{0,1,2,3,4}f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\} recursively as follows: \[f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\ f(i-1,1) & \text{ if } i \ge 1 \text{ and } j = 0 \text{, and} \\ f(i-1, f(i,j-1)) & \text{ if } i \ge 1 \text{ and } 1 \le j \le 4. \end{cases}\]What is f(2015,2)f(2015,2)?
Plain-text mathematical notation (without MathML)
For every positive integer n, let mod₅(n) be the remainder obtained when n is divided by 5. Define a function f:{0,1,2,3,…}×{0,1,2,3,4}→{0,1,2,3,4} recursively as follows:

\[f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\ f(i-1,1) & \text{ if } i \ge 1 \text{ and } j = 0 \text{, and} \\ f(i-1, f(i,j-1)) & \text{ if } i \ge 1 \text{ and } 1 \le j \le 4. \end{cases}\]What is f(2015,2)?
Original LaTeX notation
For every positive integer $n$, let $\text{mod}_5 (n)$ be the remainder obtained when $n$ is divided by 5. Define a function $f: \{0,1,2,3,\dots\} \times \{0,1,2,3,4\} \to \{0,1,2,3,4\}$ recursively as follows:

\[f(i,j) = \begin{cases}\text{mod}_5 (j+1) & \text{ if } i = 0 \text{ and } 0 \le j \le 4 \text{,}\\ f(i-1,1) & \text{ if } i \ge 1 \text{ and } j = 0 \text{, and} \\ f(i-1, f(i,j-1)) & \text{ if } i \ge 1 \text{ and } 1 \le j \le 4. \end{cases}\]What is $f(2015,2)$?

level

Level 5

type

Intermediate Algebra

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task_id
d75715d6-a824-5f9c-94f2-03352854f92b
task_key
intermediate~5falgebra--test--d75715d6-a824-5f9c-94f2-03352854f92b
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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