Number Theory

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Foxellar

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Foxellar

#1 h Mar 23, 2025, 11:43 PM

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**Problem:**

Define $s(n)$ as the sum of the digits of a natural number $n$ . Also, define
$\{x\} \text{ as the fractional part of } x.$ The smallest natural number $n$ such that
$\left\{\frac{n}{s(n)}\right\} = \frac{1}{6}$ is $\boxed{?}$ .

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HockeyMaster85

34 posts

HockeyMaster85

#2 h Mar 24, 2025, 3:34 AM

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is this not just an exercise in Python?

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aidan0626

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aidan0626

#3 h Mar 24, 2025, 3:43 AM

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HockeyMaster85 wrote:

is this not just an exercise in Python?

i believe this is an exercise in number theory

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Pal702004

522 posts

Pal702004

#4 h Monday at 8:06 AM

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$s(n)$ must be divisible by $27$ and $n$ must not be divisible by $27$ (only by 9). Also $s(n)$ must be even. So, $s(n) \ge 54$

Let's try $s(n)=54$ .

$999999$ doesn't work because it's divisible by $27$

$1899999$ work.

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HockeyMaster85

34 posts

HockeyMaster85

#5 h Monday at 10:48 PM

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aidan0626 wrote:

HockeyMaster85 wrote:

is this not just an exercise in Python?

i believe this is an exercise in number theory

I meant it looked more tedious but easy to implement in python. but yeah my code aligns with @above

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HockeyMaster85

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HockeyMaster85

#6 h Monday at 10:50 PM

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Pal702004 wrote:

$s(n)$ must be divisible by $27$ and $n$ must not be divisible by $27$ (only by 9). Also $s(n)$ must be even. So, $s(n) \ge 54$

Let's try $s(n)=54$ .

$999999$ doesn't work because it's divisible by $27$

$1899999$ work.

wait why must s(n) be divisible by 27?

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ddot1

24286 posts

ddot1

#7 h Monday at 11:48 PM • 2 Y

Y by HockeyMaster85, Foxellar

Since $\dfrac{n}{s(n)}$ has denominator $6$ , that means $s(n)$ is divisible by $3$ . But then $n$ is divisible by $3$ as well. So if we have a factor of $3$ in the denominator after simplification, $s(n)$ needs to be divisible by $9$ . But then $n$ is divisible by $9$ as well. That forces $s(n)$ to be divisible by $27$ .

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HockeyMaster85

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HockeyMaster85

#8 h Yesterday at 12:39 AM

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Oh right, that's clever, since n and s(n) have the same remainder mod 9. And the problem thankfully is simple once u realize s(n) must also be even. cool

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