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# Number Theory

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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 $\phi(2835)$?

Jun 20, 2024

#1
+1230
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We can use Euler's Totient Function.

Let's find all the distinct prime factors of 2835 first.

The prime factors of 2835 are $$3,5,7$$

Now, we simplfy do $$2835(1-1/3)(1-1/5)(1-1/7)$$

Simplfying this, we have

$$2835 * \frac{2}{3}*\frac{4}{5}*\frac{6}{7} =\frac{2835\cdot \:2\cdot \:4\cdot \:6}{1\cdot \:3\cdot \:5\cdot \:7}=1296$$

I'm not sure if I did this correctly...

Thanks! :)

Jun 20, 2024
edited by NotThatSmart  Jun 20, 2024

#1
+1230
+1

We can use Euler's Totient Function.

Let's find all the distinct prime factors of 2835 first.

The prime factors of 2835 are $$3,5,7$$

Now, we simplfy do $$2835(1-1/3)(1-1/5)(1-1/7)$$

Simplfying this, we have

$$2835 * \frac{2}{3}*\frac{4}{5}*\frac{6}{7} =\frac{2835\cdot \:2\cdot \:4\cdot \:6}{1\cdot \:3\cdot \:5\cdot \:7}=1296$$

I'm not sure if I did this correctly...

Thanks! :)

NotThatSmart Jun 20, 2024
edited by NotThatSmart  Jun 20, 2024
#2
+129725
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Good job, NTS!!!!

CPhill  Jun 20, 2024
#3
+1230
+1

Thanks to you too, CPhill.

I did not know about Euler's Totient Function until today when we did a problem similar earlier.

Sure saved me a lot of time on this one! Lol! :)

Thanks! :)

~NTS

NotThatSmart  Jun 20, 2024
edited by NotThatSmart  Jun 20, 2024