Question

The greatest number which divides \(25^{n}-24 n-1\) for all \(n \in N\) is (1) 24 (2) 578 (3) 27 (4) 576

Short answer

Answer: 24

Step-by-step solution

Analyze the given expression

The given expression is \(25^{n}-24n-1\). We need to find the greatest common divisor that divides this term for all n in the set of natural numbers.

Noticing that the expression is in the form of difference of squares

Upon carefully analyzing, we can notice that the given expression \(25^{n}-24n-1\) could be rewritten in the form of difference of squares.

Rewrite the expression

We can rewrite the expression as follows: \(25^{n}-24n-1 = (25^{\frac{n}{2}})^2 - (24\frac{n}{2} + 1)^2\)

Apply the difference of squares formula

Applying the difference of squares formula, we have: \(25^{n}-24n-1 = [(25^{\frac{n}{2}} + 24\frac{n}{2} + 1) (25^{\frac{n}{2}} - (24\frac{n}{2} + 1))]\)

Factor out the common factor

From the right-hand side of the equation above, we can notice that there is a common factor of \(24\frac{n}{2}\): \(25^{n} - 24n - 1 = 24\frac{n}{2} [(25^{\frac{n}{2}} + 1) - 1][(25^{\frac{n}{2}} - 1) + 1]\)

Determine the common factor

From step 5, the greatest common factor should be \(24\) since, for any natural number \(n\), the term \(24\frac{n}{2}\) is divisible by \(24\). So, the answer is (1) 24.

Key concepts

Natural Numbers

Natural numbers are the basic building blocks of mathematics that everyone learns from the beginning. They are the set of positive integers starting from 1 and going upwards to infinity: 1, 2, 3, and so on. An important property of natural numbers is that they are always positive. Unlike whole numbers, which include zero, natural numbers start from 1.

When working with these numbers, especially in algebraic expressions, it simplifies things because their positive nature keeps operations straightforward. In the expression given in the exercise, the variable \( n \) represents a natural number. This means that \( n \) can be any positive integer, which helps us to make certain assumptions and simplifications while solving for the greatest common divisor or any arithmetic property.

Understanding natural numbers well is key to mastering many mathematical concepts, such as arithmetic progression, series, and solving polynomial expressions like the one in our problem.

Difference of Squares

The difference of squares is a special factorization formula used in algebra. It helps simplify expressions where you have two squares being subtracted from one another. The formula is \( a^2 - b^2 = (a+b)(a-b) \). This is quite useful because it breaks down a complex expression into a product of two simpler expressions.

In our exercise, the expression \( 25^n - 24n -1 \) can be rewritten to fit this form. By identifying suitable values for \( a \) and \( b \), we can factor it effectively using the difference of squares method. This is a valuable mathematical tool that allows us to quickly and efficiently solve problems that might otherwise involve lengthy calculations.

Understanding and applying the difference of squares is crucial because it not only helps with the simplification of algebraic expressions but also in finding the greatest common divisors among other applications.

Factoring Techniques

Factoring techniques are methods used in mathematics to break down complex algebraic expressions into simpler, more manageable parts. One of the most common techniques is factoring by grouping, where you group terms in such a way that you can apply common factors.

In the exercise, the expression \( 25^n - 24n - 1 \) was transformed into a form that allowed us to apply the difference of squares method. As we apply factoring techniques, it's crucial to identify common factors in the grouped terms. For example, in our solution, recognizing that \( 24 \) is a common factor in the expression after it was rewritten was key to determining the greatest common divisor.

Mastering various factoring techniques is essential as it enables you to solve not just algebraic problems but also helps in understanding number theory, which is crucial for determining factors, multiples, and divisors of numbers. This understanding leads to effective problem solving in various mathematical contexts.