Question

For what values of \(\mathrm{n}\) is \(14^{n}+11^{"}\) divisible by \(5 ?\) (1) when \(\mathrm{n}\) is an even positive integer (2) For all values of \(\mathrm{n}\) (3) When \(\mathrm{n}\) is a prime number (4) When \(\mathrm{n}\) is a odd positive integer

Short answer

For the given expression, \(14^n + 11^n\), it is divisible by 5 when \(n\) is an odd positive integer.

Step-by-step solution

Choice (1): Even Positive Integer

First, let's try the case when \(n\) is an even positive integer. When \(n\) is even, we can rewrite \(n = 2k\) for some integer \(k\). The expression becomes: $$ 14^{2k} + 11^{2k} $$ Since \((a^b)^c = a^{bc}\), we can rewrite as: $$ (14^2)^k + (11^2)^k $$ Now, notice that \(14^2 = 196\) and \(11^2 = 121\). Using the fact that \(a \equiv b \pmod{m}\) means that \(a^k \equiv b^k \pmod{m}\) for any positive integer \(k\), we have: $$ 196^k+121^k \equiv 1^k+1^k \pmod{5} $$ Therefore, the expression is always congruent to 2 modulo 5 when \(n\) is even. This means the expression is not divisible by 5 for even positive integers. So, choice (1) is incorrect.

Choice (2): For All Values of n

Since we have already disproved that the expression is always divisible by 5 for even positive integers, choice (2) is incorrect as well.

Choice (3): Prime Number

For choice (3), we'll test a few cases where \(n\) is a prime number: - When \(n = 2\), \(\phantom{=}14^2 + 11^2\phantom{=} = 196 + 121 = 317\). Not divisible by 5. - When \(n = 3\), \(\phantom{=}14^3 + 11^3\phantom{=} = 2744 + 1331 = 4075\). Divisible by 5. - When \(n = 5\), \(\phantom{=}14^5 + 11^5\phantom{=} = 537824 + 161051 = 698875\). Divisible by 5. We see that it's inconclusive from the initial tests. Furthermore, as prime numbers include both even and odd numbers, it's unlikely to find a consistent pattern. Thus, choice (3) is likely incorrect.

Choice (4): Odd Positive Integer

Finally, let's consider the case when \(n\) is an odd positive integer. When \(n\) is odd, we can rewrite \(n=2k + 1\) for some integer \(k\). Now, let's check a few odd values for \(n\) : - When \(n = 1\), \(\phantom{=}14^1 + 11^1\phantom{=} = 14 + 11 = 25\). Divisible by 5. - When \(n = 3\), \(\phantom{=}14^3 + 11^3\phantom{=} = 2744 + 1331 = 4075\). Divisible by 5. - When \(n = 5\), \(\phantom{=}14^5 + 11^5\phantom{=} = 537824 + 161051 = 698875\). Divisible by 5. Seeing the pattern, we can analyse the general expression when \(n = 2k+1\) for some integer \(k\): $$ 14^{2k+1} + 11^{2k+1} = 14^{(1)}14^{(2k)} + 11^{(1)}11^{(2k)} $$ Notice that \(14 \equiv -1 \pmod{5}\) and \(11 \equiv 1 \pmod{5}\). Using the facts that \(a \equiv b \pmod{m} \Rightarrow ac \equiv bc \pmod{m}\) and \(a^k \equiv b^k \pmod{m}\) for any positive integer \(k\), we have: $$ (-1)(-1)^{2k} + (1)(1)^{2k} \equiv 1 \pmod{5} $$ Therefore, when \(n\) is an odd positive integer, the expression is divisible by 5. We can conclude that choice (4) is the correct answer.

Key concepts

Positive Integers

Positive integers are whole numbers greater than zero. These numbers form part of what we call the "/natural numbers," which start from 1 and continue indefinitely. Here are some key points about positive integers that are easy to remember:

• The smallest positive integer is 1.
• Positive integers do not include decimals or fractions.
• They are used extensively in counting objects, ordering, and labeling.

When looking at problems involving divisibility or powers like in the example exercise, it is important to identify whether the number in question is a positive integer. Understanding that the exercises deal with positive integers ensures that the answers fit into the required criteria in mathematical problems.

Congruent Modulo

In mathematics, congruence modulo is a powerful concept used in number theory. When we say two numbers are congruent modulo a number, it means that they have the same remainder when divided by that number. This concept is visualized via the "mod" operator.

For example, if we say that two numbers, say 14 and 4, are congruent modulo 5, we express it as 14 \( \equiv \) 4 \( \pmod{5} \). This means when 14 and 4 are each divided by 5, the remainder is the same (in this case, 4). Here are some more points to clarify congruence modulo:

• It is denoted by the symbol \( \equiv \).
• If \( a \equiv b \pmod{m} \), then \( a-b \) is a multiple of \( m \).
• It simplifies number operations, especially for large numbers.

This concept is crucial in understanding problems that involve divisibility, such as seeing whether a sum like \( 14^n + 11^n \) is divisible by 5.

Prime Numbers

Prime numbers are a fundamental aspect of mathematics. They are numbers greater than 1 with no divisors other than 1 and themselves. Prime numbers cannot be formed by multiplying any two smaller natural numbers. A few quick facts about primes include:

• The number 2 is the smallest and only even prime number.
• Prime numbers are the building blocks of all numbers because any number can be expressed as a product of primes, known as its prime factorization.
• Examples of prime numbers include 2, 3, 5, 7, 11, and so on.

When dealing with the exercise, knowing that a prime number includes both even and odd numbers is crucial. Problems involving prime numbers often explore their unique properties, like how they might influence results in equations or sequences, such as checking for divisibility based on the values of \( n \) as prime numbers.