Number System | Highest power of Prime number | Examples | Aptitude | Part- 23 | Bharath Kumar

TL;DR

This session explains calculating the highest powers of prime numbers in factorials through examples.

Transcript

hi everyone welcome to the session in this session i will explain some problems related to highest power of prime number in n factorial in the last session i have explained about the concept related to highest power of n factorial uh highest power of prime number in n factorial right now in this session i will explain the problems based on uh that ... Read More

Key Insights

• ✋ The calculation of highest powers in factorials is essential for number theory and combinatorics.
• ✋ Breaking down composite numbers into their prime factors simplifies finding their highest powers in factorial operations.
• 🧑‍🏭 Each division must be counted as necessary to capture all contributions of the prime factors.
• ✋ Summing the quotients of divisibility by primes reveals their highest occurrences within factorials.
• 💦 Understanding how to manipulate prime works can aid in broader mathematical problems.
• #️⃣ Not all numbers require the same approach; consistency in method depends on whether the numbers involved are primes or composites.
• #️⃣ Examples provided in the session illustrate the method clearly for different numbers, enhancing comprehension.

Questions & Answers

Q: Why is it important to confirm if the number is prime before proceeding with the calculations?

Confirming that the number is prime is crucial because the method for finding the highest power only applies to prime numbers. If the number is not prime, it must first be converted into its prime factorization, which affects how the factorial operation is performed and how the quotients are calculated.

Q: How do you find the highest power of a prime in n factorial?

To find the highest power of a prime in n factorial, divide n by the prime number repeatedly, summing the quotients until the result is less than 1. This sum gives the highest power of the prime number in that factorial, which accounts for how many times that prime can form factors in the factorial's product.

Q: What is the procedure for finding the highest power of a composite number in a factorial?

When dealing with a composite number, first convert it to its prime factorization. For example, with the number 8, which can be expressed as 2^3, identify the prime base and calculate the highest power of that prime in the factorial, and then adjust by the composite number's exponent in the prime factorization.

Q: In the example of finding the highest power of 3 in 150 factorial, what quotients were used?

For the calculation, 150 was divided by 3 successively, yielding quotients of 50, 16, 5, and 1. The final result is the sum of these quotients, which gives the highest power of 3 in 150 factorial as 72, after dividing by the power of the prime in the factor.

Summary & Key Takeaways

• The session focuses on determining the highest power of various prime numbers in specific factorials, including 2, 3, and 5.

• The procedure involves dividing the factorial number by the prime number successively to find the corresponding quotients, which are summed to find the highest power.

• Additionally, it discusses the conversion of composite numbers like 8 into prime factor form before applying the method to determine the highest power.