Maths Olympiad Prep

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Number theory Difficulty 5.5 AIME, harder Find the answer

29. Find the prime factorizations of
a) 106−110^{6}-1
d) 224−12^{24}-1
b) 108−110^{8}-1
e) 230−12^{30}-1
c) 215−12^{15}-1
f) 236−12^{36}-1.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

29. a) 32⋅11⋅73−101.1373^{2} \cdot 11 \cdot 73-101.137 c) 7.31⋅1517.31 \cdot 151
d) 32⋅5⋅7⋅13⋅17⋅2413^{2} \cdot 5 \cdot 7 \cdot 13 \cdot 17 \cdot 241
f) 33⋅5⋅7⋅13⋅19⋅37⋅73⋅1093^{3} \cdot 5 \cdot 7 \cdot 13 \cdot 19 \cdot 37 \cdot 73 \cdot 109
e) 52⋅13⋅41⋅61⋅13215^{2} \cdot 13 \cdot 41 \cdot 61 \cdot 1321

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.