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 $p+q.$

Imnotamaster Sep 29, 2019

#1**+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

CalculatorUser Sep 29, 2019

#2**+3 **

1) The answer is 61

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

CalculatorUser Sep 30, 2019

#3**+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.**

Guest Sep 30, 2019

#4**+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