Question

Which of the following cannot be determined? (a) Range of the factors of 64 . (b) Range of the first 10 positive integers (1) a (2) b (3) Both a and b (4) None of these

Short answer

a) The range of the factors of 64 b) The range of the first 10 positive integers c) Both a and b d) None of these Answer: d) None of these

Step-by-step solution

Find the factors of 64

To find the factors of 64, we need to find all the numbers that can be multiplied by another number to get 64. The factors of 64 are 1, 2, 4, 8, 16, 32, and 64.

Determine the range of the factors of 64

To find the range, subtract the smallest value from the largest value. In this case, the smallest factor is 1 and the largest factor is 64. So, the range of the factors of 64 is 64 - 1 = 63.

Determine the range of the first 10 positive integers

The first 10 positive integers are 1, 2, 3, 4, 5, 6, 7, 8, 9, and 10. The smallest integer is 1 and the largest integer is 10. So, the range of the first 10 positive integers is 10 - 1 = 9.

Identify which of the options cannot be determined

We have found the range for both options (a) and (b). Thus, both options can be determined. Therefore, the correct answer is (4) None of these.

Key concepts

Factors of a Number

Factors of a number are the numbers that can be multiplied together to get the original number. Simply put, if you multiply two numbers and their product equals to your original number, then they both are considered factors of that number. For example, for the number 64, both 8 and 8 are factors because when you multiply them, you get 64.

Factors are important in mathematics because they help us understand numbers better. They allow us to break down complex numbers into simpler parts. You can find factors by dividing your original number by other numbers to see if you get a whole number. If you do, then that number is a factor.

Positive Integers

Positive integers are numbers that are greater than zero and do not have any decimal or fractional parts. These are often referred to as "whole numbers" and include the numbers 1, 2, 3, and so on. They are straightforward and easy to understand because they do not include zero or negative numbers.

In various mathematical operations and problem-solving scenarios, positive integers are used for counting, ordering, and many basic mathematical functions. They are fundamental in mathematics and often appear as the starting point when learning more advanced concepts.

Finding Range

The range in a set of numbers is the difference between the largest and smallest numbers in that set. It gives us an idea of how spread out the numbers are. To find the range, simply subtract the smallest number from the largest number.

For instance, if you have the factors of 64, which are 1, 2, 4, 8, 16, 32, and 64, the smallest is 1 and the largest is 64. So, the range would be 64 - 1 = 63. The range tells you how wide or narrow the your set is.

Problem Solving in Mathematics

Problem-solving in mathematics involves a variety of techniques to find solutions to questions or puzzles. It starts with understanding the problem, identifying relevant information, and then working systematically towards a solution.

Strategies may include breaking the problem into smaller parts, using patterns to determine solutions, or applying mathematical operations and concepts like factors and ranges. For example, when given a problem involving factors of a number, it's crucial to first list out all possible factors, determine their range, and then apply that understanding to find if any further calculations or comparisons are needed.

• Break down the problem.
• Identify what you know and don't know.
• Use logical steps to deduce a solution.

These steps make mathematical problem-solving a skill that is built over time with practice and experience.