Competition problem — Number theory

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Number theory Difficulty 5.5 AIME, harder Prove it Estonia

Is there a two-digit number $n$ that does not end with zero such that

a) all numbers that can be formed by adding one or more zeros between the two-digit number's digits are its multiples?

b) none of the numbers that can be formed by adding one or more zeros between the two-digit number's digits are its multiples?

c) some numbers that can be formed by adding one or more zeros between the two-digit number's digits are its multiples and some are not?

Solution

a) One such number is $n = 15$ . All numbers formed by adding zeros between its digits are divisible by $3$ and $5$ .

b) One such number is $n = 12$ . Adding zeros between the digits, the last two digits will always be $02$ . Hence no such number will be divisible by $4$ .

c) One such number is $n = 11$ , because $101$ is not divisible by $11$ but $1001$ is.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.