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Can someone please explain this in detail?

Find the sum of the distinct prime factors of $5^5 - 5^3$.
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2 Answers

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10

Work Shown:

[tex]5^5 - 5^3\\\\5^3(5^2 - 1)\\\\5^3(25 - 1)\\\\5^3(24)\\\\5^3(8*3)\\\\5^3(2^3*3)\\\\2^3*3*5^3\\\\[/tex]

The distinct prime factors are: 2, 3, 5. We'll ignore the exponents at this point.

Those primes sum to 2+3+5 = 10
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Factorize the given number:

[tex]5^5 - 5^3 = 5^3 \left(5^2 - 1\right) = 5^3 \times 24 = 2^3 \times 3 \times 5^3[/tex]

The first equality follows from

[tex]5^5 = 5^{3 + 2} = 5^3 \times 5^2[/tex]

so both 5⁵ and 5³ share a common factor of 5³ that we can pull out as we did.

Then the distinct prime factors are 2, 3, and 5; their sum is 2 + 3 + 5 = 10.
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