Question

Find the remainder when the square of any prime number greater than 3 is divided by 6 . (1) 1 (2) 3 (3) 2 (4) 4

Short answer

Answer: 1

Step-by-step solution

List several primes greater than 3, and find their squares.

Let's list the first few prime numbers greater than 3 and find their squares: - 5 (prime): 5^2 = 25 - 7 (prime): 7^2 = 49 - 11 (prime): 11^2 = 121 - 13 (prime): 13^2 = 169

Calculate the remainders when these squares are divided by 6

Now we will divide the squares of the prime numbers by 6 and find their remainders: - 25 ÷ 6 = 4 remainder 1 - 49 ÷ 6 = 8 remainder 1 - 121 ÷ 6 = 20 remainder 1 - 169 ÷ 6 = 28 remainder 1 Notice that in all cases, the remainder is 1.

Prove that this pattern holds true for all primes greater than 3

All prime numbers greater than 3 can be written in one of the following two forms: 1) 6n + 1, where n is an integer (e.g., 5, 11, 17, ...) 2) 6n - 1, where n is an integer (e.g., 7, 13, 19, ...) Now let's consider the two cases: 1) If p = 6n + 1: p^2 = (6n + 1)^2 = 36n^2 + 12n + 1 If we divide p^2 by 6, the remainder is 1. 2) If p = 6n - 1: p^2 = (6n - 1)^2 = 36n^2 - 12n + 1 If we divide p^2 by 6, the remainder is 1. In both cases, we have shown that the remainder is always 1 when the square of a prime number greater than 3 is divided by 6. Thus, the correct answer is (1) 1.

Key concepts

Modular Arithmetic

When we talk about modular arithmetic, we are diving into a system of arithmetic for integers where we only use the remainder when dividing. It’s often called "clock arithmetic" because of its cyclic nature. For example, on a clock, after 12 comes 1, then 2, and so on, wrapping back around after 12.
A crucial part of modular arithmetic is expressing calculations where specific conditions or rules apply based on remainders. In the context of this exercise, we were interested in what happens when the square of a prime number is divided by 6. Instead of focusing on the result of the entire division, we only cared about the remainder left when we divided these squared values.

• For example, when we compute the square of 5, we get 25. Dividing 25 by 6 gives us a quotient and a remainder, which in this case is 1.
• This process repeats for other primes such as 7, 11, etc., confirming the systematic behavior predicted by modular arithmetic.

Understanding this allows us to predict patterns and behaviors without performing extensive calculations.

Remainders

Remainders are a simple yet crucial concept in mathematics that show what is left after division. When a number does not divide evenly by another, the remainder is what’s leftover.
For instance, dividing 25 by 6 results in a quotient of 4 with a remainder of 1 (because 4 complete sets of 6 are 24, leaving 1). This is because 25 is not a multiple of 6.
In our exercise, we saw that after squaring several prime numbers greater than 3 and dividing by 6, the remainder was consistently 1, no matter which prime was used. This observation suggests a pattern or rule, often seen in problems like these where we have divisibility and remainder determination.

• This outcome helps simplify complex problems by focusing merely on what’s leftover in the division, providing solutions with less computational effort.
• Identifying the remainder helps mathematicians and learners make powerful generalizations, forming the basis of many mathematical proofs.

Prime Number Properties

Prime numbers are intriguing because they are only divisible by 1 and themselves. They act as the building blocks for all numbers because every number is either a prime or a product of primes.
However, prime numbers have even more interesting properties when interacting with arithmetic rules. One unique aspect discussed here is how they behave when squared and divided by 6.
For primes greater than 3, each can be expressed in the form of either \(6n + 1\) or \(6n - 1\), where \(n\) is an integer. This expression helps illustrate consistent properties they hold. When squared, these primes always result in a number that, when divided by 6, gives a remainder of 1. This property might seem magical at first, but it is essentially a characteristic that arises from their nature in modular systems.

• These properties are not just coincidental but form an essential part of number theory.
• Understanding these characteristics aids significantly in tackling similar mathematical problems and discovering new mathematical insights.

Recognizing these consistent outcomes helps harness the utility of primes in broader mathematical contexts and proofs.