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# Without using a calculator, find the largest prime factor of 15^6 - 7^6

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Without using a calculator, find the largest prime factor of 15^6 - 7^6

Dec 22, 2017

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15^6 - 7^6 =>
(15^3 - 7^3) * (15^3 + 7^3) =>
(15 - 7) * (15^2 + 15 * 7 + 7^2) * (15 + 7) * (15^2 - 15 * 7 + 7^2) =>
8 * (225 + 105 + 49) * 22 * (225 - 105 + 49) =>
8 * 22 * (274 + 105) * (274 - 105) =>
8 * 22 * 379 * 169 =>
2^3 * 2 * 11 * 13^2 * 379 =>
2^4 * 11 * 13^2 * 379

This is not my work, but a solution on Yahoo Answers.

Dec 22, 2017
#2
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Both 15^6 and  7^6 are squares, so we can factor 15^6 - 7^6  as the difference of squares:

15^6-7^6 = (15^3-7^3)(15^3+7^3)

We can pound out the computation from here (yuck!), or we can factor 15^3-7^3  as a difference of cubes and 15^3+7^3 as a sum of cubes:

[(15^3-7^3)(15^3+7^3) = [(15-7)(15^2+15*7 + 7^2)][(15+7)(15^2-15*7 + 7^2)]

The largest of these factors is 379.

I did a better solution on paper, but I can't attach a file, sorry!

Dec 22, 2017
edited by supermanaccz  Dec 22, 2017
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Excellent, Julius and superman !!!!   Dec 22, 2017