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1)Find the largest prime factor of $3^{15} + 3^{11} + 3^{6} + 1.$

 

2)Suppose that $p$ and $q$ are prime numbers such that $p$ divides $q+1$ and $q$ divides $p+1.$ Determine $p+q.$

 Sep 29, 2019
edited by Imnotamaster  Sep 29, 2019
 #1
avatar+2417 
+3

2) The two prime numbers have to be one away from each other.

 

So it is 2 and 3.

 

Lets check:

 

2 / (3+1)

 

3/ (2+1)

 

Yes, so P + Q is 5

 Sep 29, 2019
 #2
avatar+2417 
+3

1) The answer is 61

 

I don't know why though! sad. I brute forced the problem...

 Sep 30, 2019
 #3
avatar
+2

1 - Sum them up first: 3^15 + 3^11 + 3^6 + 1=14,526,784  
Then factor the sum: 14,526,784 = 2^6×61^3 (9 prime factors, 2 distinct)
As you can see, the largest prime factor is 61.

 Sep 30, 2019
 #4
avatar+2417 
+2

That is what I did, but there has to be a trick to do this without a calculator or internet

CalculatorUser  Sep 30, 2019

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