u/Major_Tap4199

Came up with this one for fun, no idea if it's been posted before somewhere. Fair warning, I'm not amazing at math, just got curious about this one and worked through it slowly. Mostly wanted to share because I liked how a silly real-world setup ended up landing right on top of φ(n).

The setup

In an online poll, viewers vote either "Yes" or "No," and the result is displayed only as a percentage rounded to exactly two decimal places (e.g., 41.27%). The total number of votes is not shown. Assume that for any percentage displayed, the actual vote tally is the minimum possible whole number of votes that could have produced that exact percentage.

The question

Out of all possible displayed percentages (from 00.01% to 99.99% in steps of 0.01%), how many of them require the full 10,000 voters as a minimum? And which displayed percentages are those, intuitively?

>!Where coprimality comes in!<

>!A displayed percentage X.XX% corresponds to the fraction XXXX/10000. The minimum number of voters needed to produce that exact ratio is 10000 / gcd(XXXX, 10000). So the minimum hits its maximum (10,000) exactly when gcd(XXXX, 10000) = 1, i.e., when the numerator is coprime to 10,000.!<

>!Two numbers are coprime when they share no prime factors. Since 10,000 = 2⁴ × 5⁴, its only prime factors are 2 and 5. So XXXX is coprime to 10,000 if and only if XXXX is odd AND not divisible by 5. That's a clean shortcut, you don't have to actually factor the numerator at all, you just check the last digit.!<

>!Where Euler's totient comes in!<

>!The count of integers from 1 to n that are coprime to n is exactly Euler's totient function φ(n). For n = 10,000:!<

>!φ(10000) = 10000 × (1 − 1/2) × (1 − 1/5) = 10000 × 0.5 × 0.8 = 4,000!<

>!So exactly 4,000 displayed percentages require the full 10,000 voters as a minimum. That's 40% of all possible X.XX displays.!<

>!The pattern generalizes nicely. If you display to d decimal places, the max minimum is 10^(d+2), and the number of splits tied at that max is φ(10^(d+2)) = 0.4 × 10^(d+2). Always exactly 40%, because the prime factorization of any power of 10 only involves 2 and 5, and (1 − 1/2)(1 − 1/5) = 0.4.!<

>!The part I thought was nice!<

>!The reason the answer is always 40% (regardless of how many decimal places you display) is that 10 only has two prime factors. If we counted in some weird base where the denominator had more prime factors, the proportion of "hardest" splits would drop. The fact that our base-10 display gives such a clean answer is a small accident of the base we count in.!<

Curious if anyone sees a slicker way to frame the general result, or if there's a related problem I should look at. Also happy to be told this is a well-known exercise and I just reinvented it.

Came up with this the other day, no idea if it's been posted before somewhere, just thought it was a fun little problem.

Fair warning, I'm not amazing at math, just got curious about this one. Mostly wanted to see how quickly a math savvy person could crack it versus how long it took me to work through.

In an online poll, viewers vote either "Yes" or "No," and the result is displayed only as a percentage rounded to exactly two decimal places (e.g., 41.27%). The total number of votes is not shown. Assume that for any percentage displayed, the actual vote tally is the minimum possible whole number of votes that could have produced that exact percentage.

Which of the following displayed results required the greatest minimum number of voters?

(A) 31.25% Yes / 68.75% No
(B) 27.50% Yes / 72.50% No
(C) 38.46% Yes / 61.54% No
(D) 43.21% Yes / 56.79% No
(E) 47.92% Yes / 52.08% No

Curious if people find the shortcut quickly or if it takes some grinding. Answer in spoilers once you've got it, and if you don't mind, drop how long it took you.

>!!Answer: D. Once you see that 10,000 = 2⁴ × 5⁴, the only thing that matters is whether the numerator (XXXX out of 10000) shares a factor of 2 or 5. So you just check if it's odd and doesn't end in 0 or 5. 4321 passes both, gcd with 10000 is 1, minimum voters = 10,000.!!<

Hola, queria ver si alguien me puede dar info de desierto de los leones.

Quiero ir desde el ex-convento a cerro san miguel, porque creo que ahi es donde mejor se ve, no se si haya algun otra cumbre donde se pueda ver bien. Voy con alguien que no es super hiker entonces quiero saber mas o menos cuanto nos tomaria y que tan dificil esta. Igualmente si saben algo de que tan seguro es ir ahi a las 3-4 am (asumiendo 2 horas de subida). Gracias

so i kind of wanted to hide my story from 1000+ people, i went to story settings and selected 1000 people and when i clicked hide, it said there was an error after buffering for like 3 minutes, now it says my stories are not hidden from anyone, but, they are, for some, but it says 0 people, only way to fix is to hide again then unhide from each individual person i hid, which will take about 3 years if i do it that way, anybody know how i can do it faster? Now i cant even hide or unhide since insta says it limits how poeple do certain things to protect their community, wtf do i do

Hola como estan?

Quiero ir a dinamos a ver el amanecer desde hasta arriba, pero les queria preguntar que tan seguro es esa zona (para manejar ahi tipo 2-3 am) y que tan seguro es subir la ruta completamente de noche.

Seria en un sabado, he escuchado que entre semana luego puede ser que te asalten. Iria yo solo con una chica entonces siento que si somos un grupo un poco facil de asaltar (a diferencia de ir 6 hombres juntos).

Si saben algo lo apreciaria

Hola como estan?

Quiero ir a dinamos a ver el amanecer desde hasta arriba, pero les queria preguntar que tan seguro es esa zona (para manejar ahi tipo 2-3 am) y que tan seguro es subir la ruta completamente de noche.

Seria en un sabado, he escuchado que entre semana luego puede ser que te asalten. Iria yo solo con una chica entonces siento que si somos un grupo un poco facil de asaltar (a diferencia de ir 6 hombres juntos).

Si saben algo lo apreciaria