Question

Quantity \(\mathbf{A}\) The greatest odd factor of 78 Quantity \(\mathbf{B}\) The greatest prime factor of 78 a. Quantity A is greater. b. Quantity B is greater. c. The two quantities are equal d. The relationship cannot be determined from the information given.

Short answer

The correct answer is a. Quantity A is greater.

Step-by-step solution

Finding the Greatest Odd Factor of 78

To find the greatest odd factor of 78, we can start by listing all the factors of 78: 1, 2, 3, 6, 13, 26, 39, and 78. Among these, the odd factors are 1, 3, 13, and 39. Therefore, the greatest odd factor of 78 is 39.

Finding the Greatest Prime Factor of 78

To find the greatest prime factor of 78, we again look at the list of factors. The prime factors of 78 are 2, 3, and 13. Hence, the greatest prime factor of 78 is 13.

Comparing Quantities A and B

After finding that the greatest odd factor of 78 (Quantity A) is 39 and the greatest prime factor of 78 (Quantity B) is 13, we can determine that Quantity A is larger than Quantity B.

Key concepts

Understanding Factors

Factors are numbers you can multiply together to get another number. For example, when considering the number 78, its factors are all the whole numbers that divide 78 without leaving a remainder. These include numbers like 1, 2, 3, and even 39.

Understanding factors helps to break down numbers into simpler parts, and this is crucial when solving problems like those in GRE quantitative reasoning. Factors can often determine relationships between numbers, like identifying the greatest or smallest factor in particular contexts.

• For 78, factors we considered are 1, 2, 3, 6, 13, 26, 39, and 78.
• An important aspect is determining which factors fit specific criteria, like being odd or prime.

Identifying Odd Numbers

Odd numbers are integers which aren't divisible by 2. They have specific properties and appearance among integer lists. They're usually the numbers you see between multiples of 2, such as 1, 3, 5, and so on.

For the number 78, determining odd numbers involves filtering out numbers like 2, 6, and anything divisible evenly by 2. The odd factors of 78 that surfaced from our examination were 1, 3, 13, and 39. This quality of being odd helps in scenarios where you need numbers that aren't symmetrical or regular, focusing specially on non-uniform characteristics in problem-solving.

Exploring Prime Numbers

Prime numbers hold a special place in mathematics because they can only be divided by themselves and 1. They're essentially the building blocks of numbers since any number can be broken down into these prime factors.

• For example, here, prime numbers like 2, 3, and 13 divide 78 without a remainder.
• The greatest prime factor of 78 was also a focal point, which turned out to be 13.

Prime numbers help in simplifying complex numerical relationships and are often used to evaluate the efficiency of different algorithms, especially in GRE quantitative problems.

Comparing Quantities

Comparing quantities involves evaluating two numbers to determine which one holds a greater, smaller, or equal value. This concept is essential in exercises where you need to decide between different numerical characteristics.

In the exercise, Quantity A (the greatest odd factor of 78) was 39, and Quantity B (the greatest prime factor of 78) was 13.

• We compared these two numbers to conclude that Quantity A is greater than Quantity B.
• Such comparisons are fundamental in many quantitative reasoning tasks where understanding the size and value of elements is necessary.

These comparisons are a staple in analyzing numbers in exams, ensuring a deeper understanding of how values relate to one another.