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Chapter 5: Problem 1

Evaluate \(\left(\frac{3}{53}\right)\).

Short Answer

Expert verified
Answer: The simplest form of the fraction is \(\frac{3}{53}\).

Step by step solution

01

1. Find the common factors

To simplify the fraction, we need to find any common factors between the numerator and the denominator. Prime factorize the numerator and the denominator: 3: This is a prime number, so its prime factors are 3. 53: This is also a prime number, so its prime factors are 53.
02

2. Cancel out any common factors

As we can see from the prime factors of the numerator and the denominator, they have no common factors to cancel out. Thus, the fraction is already in its simplest form.
03

3. Final Answer

The simplified fraction is \(\left(\frac{3}{53}\right)\).

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Key Concepts

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

Simplifying Fractions
Simplifying fractions is a process that helps us reduce fractions to their simplest form, where the numerator and the denominator share no common factors except for 1. By simplifying a fraction, we aim to make it easier to understand and work with, especially in complex calculations.

To simplify a fraction:
  • Identify the greatest common factor (GCF) of the numerator and the denominator.
  • Divide both the numerator and the denominator by this GCF.
In the case of the fraction \(\frac{3}{53}\), since both 3 and 53 are prime numbers and have no common factors, the fraction is already in its simplest form.
Prime Numbers
Prime numbers are the building blocks of all natural numbers. A prime number is a number greater than 1 that cannot be divided by any other number except itself and 1, without leaving a remainder. Prime numbers have exactly two distinct positive divisors: 1 and the number itself.

Understanding prime numbers:
  • Prime numbers include 2, 3, 5, 7, 11, 13, 17, and so on.
  • The number 2 is the only even prime number.
  • Every other even number can be divided by 2, making them non-prime.
In our example, both 3 and 53 are prime numbers, making their prime factorization quite simple: the number itself.
Greatest Common Factor
The greatest common factor (GCF) of two numbers is the largest number that divides both of them without leaving a remainder. Finding the GCF is essential when simplifying fractions, as it helps in reducing the fraction to its simplest form.

How to determine the GCF:
  • Prime factorize both numbers.
  • Identify the common prime factors.
  • Multiply these common factors to get the GCF.
When dealing with the fraction \(\frac{3}{53}\), both numbers are prime, meaning they have no common prime factors. Consequently, the GCF is 1, which confirms that the fraction cannot be simplified further.

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Most popular questions from this chapter

Find all primitive roots modulo 25 .

Evaluate \(\left(\frac{258}{4520}\right)\).

Find a complete set of incongruent primitive roots of 19 .

Evaluate the Legendre symbol \(\left(\frac{7}{11}\right)\) by using Euler's criterion.

Using the law of quadratic reciprocity, show that if \(p\) is an odd prime, then $$ \left(\frac{3}{p}\right)=\left\\{\begin{array}{ll} 1 & \text { if } p \equiv \pm 1(\bmod 12) \\ -1 & \text { if } p \equiv \pm 5(\bmod 12) \end{array}\right. $$
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