+1206 Without using a calculator, find the largest prime divisor of \(5^{12}- 2\cdot 10^6 +2^{12}.\)
+118609
\(5^{12}-2 \cdot 10^6+2^{12}\\ =5^{12}-2\cdot 5^6 \cdot 2^6+2^{12}\\ =(5^{6})^2-2\cdot 5^6 \cdot 2^6+(2^{6})^2\\ =(5^6-2^6)^2\\ =[(5^3)^2-(2^3)^2]^2\\ =[(5^3-2^3)(5^3+2^3)]^2\\ =[(5-2)(5^2+5\cdot 2+2^2)(5+2)(5^2-5\cdot 2+2^2)]^2\\ =[3(25+10+4)\cdot 7(25-10+4)]^2\\ =[3\cdot 39\cdot 7\cdot 19]^2\\ =[3\cdot 3\cdot 13\cdot 7\cdot 19]^2\\ \)
So the highest prime divisor is 19.
+118609
\(5^{12}-2 \cdot 10^6+2^{12}\\ =5^{12}-2\cdot 5^6 \cdot 2^6+2^{12}\\ =(5^{6})^2-2\cdot 5^6 \cdot 2^6+(2^{6})^2\\ =(5^6-2^6)^2\\ =[(5^3)^2-(2^3)^2]^2\\ =[(5^3-2^3)(5^3+2^3)]^2\\ =[(5-2)(5^2+5\cdot 2+2^2)(5+2)(5^2-5\cdot 2+2^2)]^2\\ =[3(25+10+4)\cdot 7(25-10+4)]^2\\ =[3\cdot 39\cdot 7\cdot 19]^2\\ =[3\cdot 3\cdot 13\cdot 7\cdot 19]^2\\ \)
So the highest prime divisor is 19.
