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# help

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Without using a calculator, find the largest prime divisor of $$5^{12}- 2\cdot 10^6 +2^{12}.$$

Sep 22, 2018

#1
+102461
+2

$$5^{12}-2 \cdot 10^6+2^{12}\\ =5^{12}-2\cdot 5^6 \cdot 2^6+2^{12}\\ =(5^{6})^2-2\cdot 5^6 \cdot 2^6+(2^{6})^2\\ =(5^6-2^6)^2\\ =[(5^3)^2-(2^3)^2]^2\\ =[(5^3-2^3)(5^3+2^3)]^2\\ =[(5-2)(5^2+5\cdot 2+2^2)(5+2)(5^2-5\cdot 2+2^2)]^2\\ =[3(25+10+4)\cdot 7(25-10+4)]^2\\ =[3\cdot 39\cdot 7\cdot 19]^2\\ =[3\cdot 3\cdot 13\cdot 7\cdot 19]^2\\$$

So the highest prime divisor is 19.

Sep 22, 2018

#1
+102461
+2

$$5^{12}-2 \cdot 10^6+2^{12}\\ =5^{12}-2\cdot 5^6 \cdot 2^6+2^{12}\\ =(5^{6})^2-2\cdot 5^6 \cdot 2^6+(2^{6})^2\\ =(5^6-2^6)^2\\ =[(5^3)^2-(2^3)^2]^2\\ =[(5^3-2^3)(5^3+2^3)]^2\\ =[(5-2)(5^2+5\cdot 2+2^2)(5+2)(5^2-5\cdot 2+2^2)]^2\\ =[3(25+10+4)\cdot 7(25-10+4)]^2\\ =[3\cdot 39\cdot 7\cdot 19]^2\\ =[3\cdot 3\cdot 13\cdot 7\cdot 19]^2\\$$

So the highest prime divisor is 19.

Melody Sep 22, 2018
#3
+101870
+1

Nice work, Melody!!!

CPhill  Sep 22, 2018
#4
+102461
0

Thanks Chris :)

Melody  Sep 22, 2018