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Number Theory Level 4

Given two random integers a and b, what is the percent chance that they are co-prime, rounded to the nearest whole number?

(If the chance was .512, type 51)


The answer is 61.

Alex Li by Alex Li , 21, USA

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2 solutions

The probability that two random integers are coprime is 6 π 2 ≈ 0.608 = 61 % \dfrac{6}{\pi^2} \approx 0.608=61 \% to two decimal places.

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Good reproduction of https://de.wikipedia.org/wiki/Teilerfremdheit!

Andreas Wendler - 5 years, 2 months ago
Alex Li
Mar 23, 2016

If a 2 numbers is co-prime, they are not both divisible by any primes. We can approximate this answer by finding the chance that they are not both divisible by the first couple of primes.

Chance that 2 numbers are both divisible by prime p: (1/p)^2

Chance that 2 numbers are NOT both divisible by prime p: (1 - (1/p)^2)

P(a and b) = a*b

Chance that 2 numbers are NOT both divisible by any prime:

(1 - (1/2)^2) * (1 - (1/3)^2) * (1 - (1/5)^2) * (1 - (1/7)^2) * (1 - (1/11)^2)... = ~.39

1-.39 = ~.61 = 61%

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