Problem 101 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The product of \(r\) consecutive positive integers is divisible by (A) \(r !\) (B) \((r-1) !\) (C) \((r+1) !\) (D) None of these

Step-by-Step Solution

Answer

The product is divisible by \( r! \) (Option A).

Step 1: Understand the Problem

We need to find the product of \( r \) consecutive positive integers and decide which of the given factorial terms divides this product.

Step 2: Identify the Integers

Consider \( r \) consecutive integers: \( n, n+1, n+2, \ldots, n+r-1 \). The product is \( n(n+1)(n+2)\ldots(n+r-1) \).

Step 3: Apply Factorial Division Rule

Recall that according to the factorial division rule, any set of \( r \) consecutive positive integers contains each number from 1 to \( r \) as a factor at least once. Therefore, their product is divisible by \( r! \).

Step 4: Verify Divisibility by Options

We verify for each option. For option (A), the set of \( r \) numbers indeed includes each number from 1 to \( r \), so the product is divisible by \( r! \).

Step 5: Analyze Incorrect Options

Option (B) \((r-1)!\) and option (C) \((r+1)!\) do not hold as the sets in each case either contain more numbers than necessary or miss out some factors, thereby making them incorrect choices.

Step 6: Conclusion

Based on the steps above, the correct answer is that the product of \( r \) consecutive positive integers is divisible by \( r! \).

Key Concepts

Factorial Division

When approaching problems involving factorials and division, it is essential to understand the role of factorial numbers. A factorial, denoted as \( n! \), is the product of all positive integers up to \( n \). For instance, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \). Factorial division involves checking whether a product of numbers is divisible by a factorial, which often comes up when discussing sets of numbers or sequences.

• Factorials grow very quickly, so they ensure that even larger sequences are elements of the product.
• In many mathematical problems, leveraging factorials can simplify the analysis of sequences and combinations.

In essence, factorial division is a powerful tool in combinatorics, making it possible to determine divisibility properties instantly by understanding the structure of multiplying sequences.

Consecutive Integers

Consecutive integers are numbers that follow each other in order without any gaps. For example, \(3, 4, 5\) are consecutive integers. When analyzing such numbers in mathematics, it is crucial to note their inherent properties:

• Every consecutive set of \( r \) numbers will, inherently, cover the divisions by all numbers from 1 to \( r \).
• They naturally contain a good distribution of factors, enabling products of these numbers to often be divisible by factorials.

This concept is especially vital when dealing with combinatorial proofs or problems that require verification of divisibility, such as the product of consecutive integers being divisible by their length factorial, \( r! \).
Simply put, consecutive integers preserve order and balance in sequences, making them a key player in formulation and verification.

Factorial Divisibility

The concept of factorial divisibility comes into play predominantly in combinatorics. It revolves around understanding when and why a set of numbers are divisible by a factorial, \( r! \). This notion leverages the basic principle that factorials account for all permutations and combinations up to any number \( r \).

• Within any set of \( r \) numbers, each number divided by its factorial \( r! \), ensures that every subset permutation is covered.
• The assurance of complete division by factorial stems from the fact that factorials incorporate all smaller integers as factors.

In the exercise mentioned, you'll find that the set of \( r \) consecutive numbers perfectly fits the definition for divisibility by \( r! \), due to each number appearing as a factor. This underpins a reliable approach to proving or disproving divisibility claims in factorial-related problems.