How many positive integers $k$ between $1$ and $100$ are such that $\frac{7}{k+7}$ have the form $\sum_{i=0}^n\frac{a_i}{15^i}$ with $a_i\in\Bbb{I}$?

Asked Feb 12 '22 at 15:22

Active Feb 13 '22 at 05:19

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Let a fraction that can be expressed by the sum $\;\sum_{i=0}^n\frac{a_i}{15^i}$, where $a_i \in \Bbb I$ is a $15$-based fraction.

Then find how many positive integer $k$ between $1$ and $100$ are there such that $\frac{7}{k+7}$ is a $15$-based fraction. (Source: IJSO first round test.)

I tried letting $a_j$ be the smallest number in $\{a_1,a_2,...,a_k\}$ and let $a_l$ be the largest number in $\{a_1,a_2,...,a_k\}$, to create an inequality, but that isn't working.

edited Feb 13 '22 at 05:19

asked Feb 12 '22 at 15:22

Monai

What is $\Bbb I$? – Arthur Feb 12 '22 at 15:23
You need $\frac 7{n+7}$ to terminate within $n$ terms in base $15$, so $n+7$ must divide $15^n$ – Ross Millikan Feb 12 '22 at 15:43
@Arthur $\Bbb I$ is a set of integer I guess. – Monai Feb 13 '22 at 05:23

How many positive integers $k$ between $1$ and $100$ are such that $\frac{7}{k+7}$ have the form $\sum_{i=0}^n\frac{a_i}{15^i}$ with $a_i\in\Bbb{I}$?

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