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Berkeley Math Circle

Algebra Difficulty 4.0 AMC 10/12 Find the answer United States

Problem:

Compute the prime factorization of 253−27225^{3}-27^{2}.

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

Solution

Solution:

Noticing that 25=5225=5^{2} and 27=3327=3^{3}, we have this is
56−36=(53−33)(53+33)=(5−3)(52+5⋅3+32)(5+3)(52−5⋅3+32)=2⋅49⋅8⋅19=24⋅72⋅19. \begin{aligned} 5^{6}-3^{6} &=\left(5^{3}-3^{3}\right)\left(5^{3}+3^{3}\right) \\ & =(5-3)\left(5^{2}+5 \cdot 3+3^{2}\right)(5+3)\left(5^{2}-5 \cdot 3+3^{2}\right) \\ & =2 \cdot 49 \cdot 8 \cdot 19 \\ & =2^{4} \cdot 7^{2} \cdot 19 . \end{aligned}

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.