Question

If \(n\) is a natural number, then \(9^{2 n}-4^{2 n}\) is always divisible by (1) 5 (2) 13 (3) both (1) and (2) (4) neither (1) nor (2)

Short answer

Answer: The expression \(9^{2 n} - 4^{2 n}\) is always divisible by both 5 and 13 for any natural number n.

Step-by-step solution

Factor the expression using the difference of squares method

We have the expression \(9^{2 n}-4^{2 n}\). Notice that this is in the format of \(a^2 - b^2\), with \(a = 9^n\) and \(b = 4^n\). Using the difference of squares factorization, we can rewrite the expression as \((a+b)(a-b)\): $$(9^n + 4^n)(9^n - 4^n)$$

Check for divisibility by 5

To check if the expression is divisible by 5, we need to analyze the factors separately. We know that \(9^n\) will always have a remainder of 4 when divided by 5 (since \(9 \equiv -1 \pmod{5}\)). Therefore, \(9^n \equiv -1\pmod{5}\). Similarly, \(4^n\) will always have a remainder of 4 when divided by 5 (since \(4 \equiv -1 \pmod{5}\)). Therefore, \(4^n \equiv -1\pmod{5}\). Adding the remainders, we get: $$9^n + 4^n \equiv -1 + (-1) \equiv -2 \equiv 3 \pmod{5}$$ Hence, the factor \((9^n + 4^n)\) is not divisible by 5. However, we know that the other factor, \((9^n - 4^n)\), is divisible by 5 because the remainders will cancel out: $$9^n - 4^n \equiv -1 - (-1) \equiv 0 \pmod{5}$$ Therefore, the entire expression \(9^{2 n}-4^{2 n}\) is divisible by 5.

Check for divisibility by 13

We will use a similar approach for checking divisibility by 13. Firstly, we can notice that \(9^n\) will always have a remainder of (-4) when divided by 13 (since \(9 \equiv -4 \pmod{13}\)). Therefore, \(9^n \equiv -4\pmod{13}\). Similarly, \(4^n\) will always have a remainder of 4 when divided by 13 (since \(4 \equiv 4 \pmod{13}\)). Therefore, \(4^n \equiv 4\pmod{13}\). Adding the remainders, we get: $$9^n + 4^n \equiv -4 + 4 \equiv 0 \pmod{13}$$ This means that the factor \((9^n + 4^n)\) is divisible by 13. For the other factor, subtracting the remainders, we have: $$9^n - 4^n \equiv -4 - 4 \equiv -8 \equiv 5 \pmod{13}$$ Therefore, the factor \((9^n - 4^n)\) is not divisible by 13. However, the entire expression \(9^{2 n}-4^{2 n}\) is still divisible by 13 because one of its factors, \((9^n + 4^n)\), is divisible by 13.

Determine the correct answer

Based on our analysis in steps 2 and 3, we conclude that the expression \(9^{2 n}-4^{2 n}\) is divisible by both 5 and 13. Therefore, the correct answer is: (3) both (1) and (2)

Key concepts

Difference of Squares

Understanding the difference of squares is crucial for tackling a variety of problems in algebra and number theory. It refers to an expression of the form \(a^2 - b^2\), which can be factored into \((a + b)(a - b)\). This identity is particularly handy when you deal with large numbers or expressions involving exponentiation, as it simplifies the problem by breaking it down into two smaller factors. For example, in our exercise, the expression \(9^{2n} - 4^{2n}\) represents a difference of squares, where \(a = 9^n\) and \(b = 4^n\). By applying this method, we can analyze the divisibility of each factor separately, greatly simplifying our task.

When the exercise calls for evaluating divisibility, factoring the expression first can reveal factors that have known divisibility properties, which ultimately lead to the solution. It's a powerful tool that, when mastered, serves to make complex problems much more manageable.

Number Theory

Number theory, a branch of pure mathematics, is primarily concerned with the properties of numbers, especially integers. It has applications in cryptography, computer science, and even in solving divisibility problems like the one we see in our exercise. An understanding of number theory concepts such as prime numbers, greatest common divisors, and modular arithmetic, to name a few, can provide a strong foundation for analyzing problems involving integers.

The exercise we are looking at delves into divisibility—a core number theory topic. Divisibility rules help us determine whether a number is a multiple of another without performing complete division. In our case, we explore the divisibility by 5 and 13, which follows through by understanding the behavior of powers modulo these numbers. It's this marriage of theory and practice that makes number theory both fascinating and essential in mathematics.

Modular Arithmetic

Modular arithmetic is sometimes known as 'clock arithmetic' because it involves a system where numbers wrap around upon reaching a certain value (the modulus). It's a key part of number theory and is used to solve divisibility problems among other things. For instance, saying \(9^n \equiv -1\) (mod 5), as seen in the solution for our exercise, means that when you divide \(9^n\) by 5, the remainder is the same as when you divide -1 by 5.

Using modular arithmetic simplifies calculations as it allows you to work with remainders rather than the whole numbers themselves. In the given problem, by analyzing each term of \(9^{2n} - 4^{2n}\) under modulo 5 and 13 separately, we can establish the divisibility without direct computation of potentially large numbers, which can be immensely helpful in more complex mathematical proofs or computations.

Exponentiation

Exponentiation is a mathematical operation that involves raising a number, known as the base, to the power of an exponent. In simpler terms, it tells you how many times to multiply the base by itself. It plays a pivotal role in our exercise, where we encounter expressions like \(9^{2n}\) and \(4^{2n}\). When combined with the rules of modular arithmetic, exponentiating numbers can reveal repetitive patterns in the remainders, making it clear whether a number is divisible by another.

For example, in the given exercise, noting that the remainder of \(9^n\) when divided by 5 is consistently -1 irrespective of the value of \(n\) highlights a periodicity in the remainders that exponentiation can produce. Similarly, understanding the exponents in the context of modular arithmetic is an essential skill for proving divisibility, and more broadly, it is indispensable in fields ranging from pure mathematics to computer science and cryptography.