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OlympiadBench / 1709 / For each positive integer k, let t(k) be the largest odd divisor of k.…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

question

For each positive integer kk, let t(k)t(k) be the largest odd divisor of kk. Determine all positive integers aa for which there exists a positive integer nn such that all the differences t(n+a)−t(n),t(n+a+1)−t(n+1),…,t(n+2a−1)−t(n+a−1) t(n+a)-t(n), \quad t(n+a+1)-t(n+1), \quad \ldots, \quad t(n+2 a-1)-t(n+a-1) are divisible by 4 .
Plain-text mathematical notation (without MathML)
For each positive integer k, let t(k) be the largest odd divisor of k. Determine all positive integers a for which there exists a positive integer n such that all the differences

t(n+a)−t(n), t(n+a+1)−t(n+1), …, t(n+2a−1)−t(n+a−1)

are divisible by 4 .
Original LaTeX notation
For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k$. Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences

$$
t(n+a)-t(n), \quad t(n+a+1)-t(n+1), \quad \ldots, \quad t(n+2 a-1)-t(n+a-1)
$$

are divisible by 4 .

answer type

Numerical

is multiple answer

true

language

English

question type

Open-ended

subject

Math

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Source and history

upstream_id / Original problem ID
1709
task_id
5068f874-ba3a-52b3-af8d-0a4ee4c86d89
task_key
OE~5fTO~5fmaths~5fen~5fCOMP--train--1709
task_revision_id
4
config / split
OE_TO_maths_en_COMP / train

Official source

initial import