Number Theory

 

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

 

 23  =  ends with a 3    

5¹⁹  =  ends with a 5    

7¹³  =  ends with a 7     (see below)

 

Whatever the sum of the three numbers is, it ends with a 5.    

Since the sum ends with a 5, the smallest prime divisor is 5.    

 

7⁰ = 1         7⁴ = 2,401         7⁸ = 5,764,801    

7¹ = 7         7⁵ = 16,807       7⁹ = 40,353,607    

7² = 49       7⁶ = 117,649     7¹⁰ = 282,475,249    

7³ = 373     7⁷ = 823,543     7¹¹ = 1,977,326,743    

 

In the value of 7ⁿ as n is raised to successive exponents, the     

units digit cycles 1, 7, 9, 3, repeatedly, so 7¹³ ends with a 7.    

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