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avatar+1981 

For a positive integer n, \phi(n) denotes the number of positive integers less than or equal to $n$ that are relatively prime to $n$. What is $\phi(20)$?

 Jun 20, 2024
 #1
avatar+1230 
+1

Relatively prime means when two numbers only have 1 as their common factor. 

We want numbers under 20 that share NO common divisors other than 1. 

 

Any number that is a factor of 20 won't work, since they are divisble by themselves. 1 doesn't count. This excludes

\(2, 4, 5, 10, 20\)

 

Any even number also won't work since they are both divisble by 2. This excludes

\(6,8,12,14,16,18,20\)

 

Lastly, we exclude any numbers divible by 5. This excludes

\(5,10,15\)

 

This leaves

\(1,3,7,9,11,13,17,19\)

 

There are 8 numbers. 

So 8 is our answer. 

 

Thanks! :)

 Jun 20, 2024
 #2
avatar+129725 
+1

Thanks, NTS!!!

 

Another way  using Euler's Totient Function

 

distinct prime  factors of 20 =   2, 5

 

ø(20)  =    20 ( 1 - 1/2) (1 -1/5)  =  20 (1/2)(4/5)  = 10 * 4/5  =  8

 

cool cool cool

 Jun 20, 2024

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