91Ó°ÊÓ

Prove that if \(p\) is a prime number and \(r\) is an integer with \(0

Short Answer

Expert verified

Using the factorial definition of the binomial coefficient, we have \(\left(\begin{array}{l}p \\\ r\end{array}\right) = \frac{p!}{r!(p-r)!}\). Simplifying this expression as \(\left(\begin{array}{l}p \\\ r\end{array}\right) = \frac{(p - r + 1)(p - r + 2) \dotsm p}{1 \cdot 2 \cdot 3 \dotsm r}\), we obtain an integer since \(p\) is a prime number and \(1<r<p\) ensures that the denominator will never have factors of \(p\). Therefore, the binomial coefficient \(\left(\begin{array}{l}p \\\ r\end{array}\right)\) is divisible by \(p\).

Step by step solution

01

Write down the factorial definition of binomial coefficient

Recall the definition of a binomial coefficient, and write down the expression for \(\left(\begin{array}{l}p \\\ r\end{array}\right)\) in terms of factorials: \[\left(\begin{array}{l}p \\\ r\end{array}\right) = \frac{p!}{r!(p-r)!}\]

02

Rewrite the expression as a product

Rewrite the expression as a product, using integers from 1 to \(p\) in the numerator. Note that the denominator consists of two factorials: \(r!\) and \((p-r)!\). So, we can rewrite the expression as: \[\left(\begin{array}{l}p \\\ r\end{array}\right) = \frac{1 \cdot 2 \cdot 3 \dotsm (p - r + 1)(p - r + 2) \dotsm p}{1 \cdot 2 \cdot 3 \dotsm r \cdot 1 \cdot 2 \cdot 3 \dotsm (p - r)}\]

03

Simplify the expression and get an integer

Now, let's divide the numerator by the denominator and remove common factors whenever possible: \[\left(\begin{array}{l}p \\\ r\end{array}\right) = \frac{(p - r + 1)(p - r + 2) \dotsm p}{1 \cdot 2 \cdot 3 \dotsm r}\] Since \(p\) is a prime number and \(1<r<p\), none of the factors in the denominator have \(p\) as a divisor. Therefore, the simplification will leave a product of integers with \(p\) being a factor in the numerator, and no factors of \(p\) in the denominator: \[\left(\begin{array}{l}p \\\ r\end{array}\right) = k\] for some integer \(k\). Thus, the binomial coefficient \(\left(\begin{array}{l}p \\\ r\end{array}\right)\) must be divisible by \(p\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Prime Number

Prime numbers are the building blocks of number theory. They are natural numbers greater than 1 that have no divisors other than 1 and themselves. Some examples of prime numbers include 2, 3, 5, 7, 11, and 13.
What makes prime numbers unique is their "indivisibility." This means if you try to divide a prime number by any natural number other than 1 or itself, you will always have a remainder.

• A prime number cannot be expressed as a product of two smaller natural numbers.
• In many mathematical proofs, prime numbers are essential in ensuring certain properties hold.
• They play a crucial role in cryptography, which is used to encrypt and secure data.

In the context of the given problem, knowing that the number is prime allows us to harness its properties effectively, ensuring that it is a divisor of certain expressions, like the binomial coefficient.

Factorial

Factorials are a vital part of mathematics, especially in permutations, combinations, and calculus. The factorial of a non-negative integer, denoted as \(n!\), is the product of all positive integers up to \(n\). For example, \(5! = 5 \times 4 \times 3 \times 2 \times 1 = 120\). The factorial function grows very quickly as \(n\) increases.

• The expression for a factorial can be written as: \( n! = n \times (n-1) \times (n-2) \times \,\cdots\,\times 2 \times 1 \)
• Factorials are key to computing binomial coefficients, which represent the number of ways to choose a subset from a larger set.
• The value \(0!\) is defined to be 1, which is a crucial convention for many mathematical formulas.

When dealing with binomial coefficients, as shown in the given exercise, factorials are used extensively to determine combinations, which leads us to simplifications involving divisors such as prime numbers.

Divisibility

Divisibility refers to the idea that one integer can be divided by another integer without leaving a remainder. This concept is fundamental in number theory and underpins many important mathematical theorems.
Consider an integer \(a\) is divisible by another integer \(b\) if there exists an integer \(k\) such that \(a = b \times k\). For example, 20 is divisible by 5 because \(20 = 5 \times 4\).

• Knowing whether a number is divisible by another helps simplify expressions and solve equations.
• Divisibility rules offer quick shortcuts, such as checking if a number is even to determine divisibility by 2.
• In proofs, establishing divisibility can confirm larger properties about numbers.

In the problem at hand, determining that a binomial coefficient is divisible by a prime number is an essential step and allows us to draw conclusions about the structure of binomial expansions.

Integer

Integers are numbers that have no fractional or decimal part. They include positive numbers, negative numbers, and zero. Examples of integers include -3, 0, 1, and 42.
Integers form one of the most basic and essential sets of numbers in mathematics, and they are used in nearly all fields of math, from algebra to advanced number theory.

• Integers can be added, subtracted, multiplied, and divided (except by zero).
• They are closed under addition, subtraction, and multiplication, meaning any combination of these operations will yield another integer.
• Integers are useful for counting and ordering objects.

In the context of the given exercise, integers appear in both the numerator and denominator of expressions for binomial coefficients. The entire exercise hinges on proving that the resulting value is a specific integer that has desired divisibility properties.