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What is the smallest prime divisor of $5^{19} + 7^{13} + 23$?

 Jun 22, 2024
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5^19 + 7^13 + 23 

 

5 raised to any power will end in 5

 

7^1  = 7

7^2 = 49

7^3  = 343

7^4 = 2401

7^13  ends in 7

 

So

 

5^19 + 7^13 + 23    will end in   5 + 7 + 3 =  5 

 

The smallest  prime divisor wil either be 3  or 5

 

[5^19 + 7^13 + 23] mod 3  = 2

 

[5^19 + 7613 + 23 ] mod 5 = 0

 

The smallest prime divisor is 5

 

cool cool cool

 Jun 22, 2024

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