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guddo
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How many different positive prime factors of 5^20 + 5^17 are there? 04 Dec 2023, 09:53
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25% (medium)

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72% (01:04) correct 28% (01:10) wrong based on 540 sessions

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How many different positive prime factors of 5^20 + 5^17 are there?

A. One
B. Two
C. Three
D. Four
E. More than four

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How many different positive prime factors of 5^20 + 5^17 are there? 04 Dec 2023, 09:56
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How many different positive prime factors of 5^20 + 5^17 are there?

A. One
B. Two
C. Three
D. Four
E. More than four

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5^20 + 5^17 =

= 5^17(5^3 + 1) =

= 5^17*126 =

= 5^17*2*3^2*7

Therefore, the number of different prime factors is four: 2, 3, 5, and 7.

Answer: D.
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How many different positive prime factors of 5^20 + 5^17 are there? 06 Dec 2023, 07:18
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Asked: How many different positive prime factors of 5^20 + 5^17 are there?

5^20 + 5^17 = 5^17 (5^3+1) = 126*5^17 = 2*3^2*7*5^17

Prime factors = {2,3,5,7}: 4 different positive prime factors

IMO D
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How many different positive prime factors of 5^20 + 5^17 are there? 08 Oct 2025, 06:06
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This is one of those problems where the key is recognizing that you don't want to calculate those massive powers of 5 directly. Let me walk you through the smart approach.

Here's how to think about this:

Step 1: Look for what's common

When you see \(5^{20} + 5^{17}\), notice that both terms share \(5^{17}\) as a factor. Let's pull that out:

\(5^{20} + 5^{17} = 5^{17} \cdot 5^3 + 5^{17} \cdot 1\)

\(= 5^{17}(5^3 + 1)\)

\(= 5^{17}(125 + 1)\)

\(= 5^{17} \cdot 126\)

Much better! Now you're working with manageable numbers.

Step 2: Factor 126 completely

Let's break down 126 into its prime factors:

  • 126 ÷ 2 = 63 → so 2 is a prime factor
  • 63 ÷ 3 = 21 → so 3 is a prime factor
  • 21 ÷ 3 = 7 → 3 appears again (but we only count distinct primes)
  • 7 ÷ 7 = 1 → so 7 is a prime factor

Therefore: \(126 = 2 \times 3^2 \times 7\)

Step 3: Identify all distinct primes

The complete factorization is: \(5^{17} \times 2 \times 3^2 \times 7\)

Now count the distinct prime factors:
  • 5 (from the \(5^{17}\) term - don't forget this one!)
  • 2 (from factoring 126)
  • 3 (from factoring 126)
  • 7 (from factoring 126)

That's 4 different positive prime factors.

Answer: D

The critical insight here is factoring out the common term before doing any calculations. Without that move, you'd be stuck trying to work with astronomically large numbers!

For a systematic understanding of how to approach these prime factorization problems—including the common traps students fall into (like forgetting to count 5 or counting repeated factors instead of distinct ones)—you can check out the step-by-step solution on Neuron by e-GMAT. You can also explore detailed solutions for other GMAT official questions with comprehensive frameworks and pattern recognition strategies here.

Hope this helps!
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