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Without using a calculator, find the largest prime divisor of 5^12 minus 2 times 10^6 plus 2^12.

Jul 14, 2020

#1
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The greatest prime divisor of 5^12 is 5 .

5 - 2 = 3 x 1,000,000 =3,000,000.

2 x  2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 =4,096

3,000,000 + 4,096 =3,004,096

Jul 14, 2020
#2
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The number is 5^12 minus 2 times 10^6 plus 2^12, Guest.

Nice try, though :D

Jul 14, 2020
#3
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the greatest prime factor of 5^12 is 5 because the other ones last digit is 5 and since the last digit is 5 then it is a multiple of 5 but 5 is a prime that ends up with 5 so the greatest prime divisor of 5^12 is 5 . 5 * (10^6) = 2000000 and 2000000 + 2^12 = 2004096 so the answer is 2004096.

Jul 14, 2020
#4
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Thank you very much! However, I don't think that is a prime number :(

Jul 14, 2020
#5
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Without using a calculator, find the largest prime divisor of $$5^{12}$$ minus 2 times $$10^6$$ plus $$2^{12}$$.

$$\begin{array}{|rcll|} \hline && 5^{12} - 2* 10^6 + 2^{12} \\ &=& 5^{12} - 2* \left(5*2\right)^6 + 2^{12} \\ &=& 5^{12} - 2* 5^6*2^6 + 2^{12} \\ &=& \left(5^{6}- 2^{6} \right)^2 \\ &=& \Big( (5^{3}- 2^{3}) (5^{3}+ 2^{3}) \Big)^2 \\ &=& \Big( (125-8) (125+8) \Big)^2 \\ &=& \left( 117*133 \right)^2 \\ &=& \Big( (3^2*13)*(7*19) \Big)^2 \\ \hline \end{array}$$

The largest prime divisor is $$\mathbf{19}$$ Jul 15, 2020