text: # Solving Problems From The Back - 5
  
  Breadcrumb 5B: We want to show that there is some $$L$$ such that $$p \mid 2^{ 2^n + Lp} -2^n = 2^n ( 2^{ Lp + 2^n - n } -1)$$.
  
  Breadcrumb 6B: Take $$L = n - 2^n$$. If we can apply Fermat's Little Theorem, then we are done, since $$2^{Lp} \equiv 2^{L} \equiv 2^{n - 2^n} \pmod{p}$$.
  
  Exercise 9: Can we apply Fermat's Little Theorem directly? If no, why not?
  
  Ponder this, then move on to the next note in this set.
  
  Note by Calvin Lin
  3 years, 10 months ago
  
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  2 \times 3 $$2 \times 3$$
  2^{34} $$2^{34}$$
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  \boxed{123} $$\boxed{123}$$
explanation: The text extract is a part of a mathematical problem-solving guide. It discusses a specific problem involving modular arithmetic and Fermat's Little Theorem. The extract contains a question (Exercise 9) asking whether Fermat's Little Theorem can be applied directly, and it prompts the reader to think about the problem before moving on to the next note.
distilabel_metadata.content: Evaluate the user given text extract and determine if it contains a math question and optionally and answer to this question.
  
  Output rules:
  
  1. Briefly describe the document and mention if it contains a question or a question and answer.
  2. Finalize the output with: Label: no-question,question,question-and-answer,multiple-question,multiple-question-and-answer
distilabel_metadata.role: system
distilabel_metadata.content: # Solving Problems From The Back - 5
  
  Breadcrumb 5B: We want to show that there is some $$L$$ such that $$p \mid 2^{ 2^n + Lp} -2^n = 2^n ( 2^{ Lp + 2^n - n } -1)$$.
  
  Breadcrumb 6B: Take $$L = n - 2^n$$. If we can apply Fermat's Little Theorem, then we are done, since $$2^{Lp} \equiv 2^{L} \equiv 2^{n - 2^n} \pmod{p}$$.
  
  Exercise 9: Can we apply Fermat's Little Theorem directly? If no, why not?
  
  Ponder this, then move on to the next note in this set.
  
  Note by Calvin Lin
  3 years, 10 months ago
  
  MarkdownAppears as
  *italics* or _italics_ italics
  **bold** or __bold__ bold
  - bulleted- list
  • bulleted
  • list
  1. numbered2. list
  1. numbered
  2. list
  Note: you must add a full line of space before and after lists for them to show up correctly
  paragraph 1paragraph 2
  
  paragraph 1
  
  paragraph 2
  
  [example link](https://brilliant.org)example link
  > This is a quote
  This is a quote
      # I indented these lines
  # 4 spaces, and now they show
  # up as a code block.
  
  print "hello world"
  # I indented these lines
  # 4 spaces, and now they show
  # up as a code block.
  
  print "hello world"
  MathAppears as
  Remember to wrap math in $$...$$ or $...$ to ensure proper formatting.
  2 \times 3 $$2 \times 3$$
  2^{34} $$2^{34}$$
  a_{i-1} $$a_{i-1}$$
  \frac{2}{3} $$\frac{2}{3}$$
  \sqrt{2} $$\sqrt{2}$$
  \sum_{i=1}^3 $$\sum_{i=1}^3$$
  \sin \theta $$\sin \theta$$
  \boxed{123} $$\boxed{123}$$
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The text extract is a part of a mathematical problem-solving guide. It discusses a specific problem involving modular arithmetic and Fermat's Little Theorem. The extract contains a question (Exercise 9) asking whether Fermat's Little Theorem can be applied directly, and it prompts the reader to think about the problem before moving on to the next note.
  
  Label: question