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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.

answer type

Numerical

is multiple answer

true

language

English

question

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 .

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 .

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: 1
config / split: OE_TO_maths_en_COMP / train

Official source

initial import