Lecture Notes on Factorials and Number Systems

Introduction

• Lecture by Ray Prakash
• Focus on factorials and number systems, specifically dealing with trailing zeros and highest powers of numbers dividing factorials.

Highest Power of a Number Dividing a Factorial

• Example Question: What is the highest power of 12 that divides 80 factorial?
  • Solution:
    • 12 is equivalent to (2^2 \times 3).
    • Determine how many times 2 and 3 appear in the prime factorization of 80 factorial.
    • Factor 80 factorial for 2s: (2^{78})
    • Factor 80 factorial for 3s: (3^{36})
    • Highest power of 12: 12 is (2^2 \times 3), so the minimum of (\frac{78}{2}) and (36) gives us 36.
    • Answer: (12^{36}).

Highest Power of 11 in Quotients of Factorials

• Example Question: What is the highest power of 11 in (1000! / 500!)?
  • Solution:
    • Calculate power of 11 in 1000! as 98 and in 500! as 49.
    • Subtract to find 11's power in the quotient: (11^{49}).

Challenging Factorial Problem

• Question: Find the value of (a) such that (2419! = 504^a \times B), where B is not a multiple of 7.
  • Solution:
    • Break down 504 into its factors: 7, 8, 9.
    • Determine how many times 7 appears in 2419 factorial: power is 402.
    • Since B is not a multiple of 7, equate the powers of 7 on both sides.
    • Result: (a = 402).

Concept of Skipping Zeros in Factorials

• Trailing Zeros: Determined by the number of 5s (as 10 = 2 x 5).
  • Example: From 1 to 4 factorial – no trailing zeros; from 5 factorial to 9 factorial – one trailing zero.
• Skipping Zeros: Occurs when multiple powers of 5 are added at multiples of 25.
  • E.g., 25 factorial adds two zeros where previously 1 zero was present.

Advanced Skipping Zeros Problem

• Concept: Factorials can skip zeros when multiple 5s are introduced, e.g., 625 factorial.
• Example Question: Find the minimum (K) such that no factorial has K, K+1, or K+2 trailing zeros.
  • Solution:
    • Such gaps occur when three numbers are skipped at multiples of 625 due to (5^4).
    • Calculate as: First occurs at (625!), which causes skips from 153 to 155.
    • Answer: The minimum K is 153.
  • Note: This pattern repeats at multiples of 625.

Summary

• Understanding the distribution of zeros in factorials and how they relate to powers of numbers is crucial for these types of mathematical problems.
• Use prime factorization and properties of numbers to solve complex factorial questions.