Problem 49 - Fundamentals of Mathematics

Question

Find all the factors of each number. 11

Step-by-Step Solution

Answer

The factors of 11 are 1 and 11.

Step 1: Understand Factors

When we talk about finding the factors of a number, we're looking for all whole numbers that can be multiplied by another whole number to get the original number. A factor is a divisor that divides into another number without leaving a remainder.

Step 2: Start with the Smallest Numbers

Begin testing divisibility for the smallest numbers. The smallest factor (other than 1) of any number is usually checked starting from 2. Since 11 is not divisible by 2, check the next smallest factor.

Step 3: Check Divisibility by Successive Integers

Continue testing divisibility by successive integers. For 11, check if it is divisible by 3, 4, 5, and so on, until reaching a number larger than its square root. No number other than 1 and 11 divides 11 exactly.

Step 4: Conclude with Prime Factorization

Since 11 is a prime number, its only factors are 1 and itself, 11. A prime number is only divisible by 1 and itself, so it cannot be broken down further.

Key Concepts

Divisibility

When we discuss divisibility, we are figuring out if one number can be divided by another without any remainder left over. This is when you can perfectly split a number into equal parts with no leftovers. For example, to see if 6 is a factor of 12, divide 12 by 6 to get 2, a whole number with no remainder, indicating 6 is a factor of 12. This is simply what divisibility is about.

• A number is divisible by another if the division results in a whole number.
• To test if a number is a factor, check if the division leaves a remainder of zero.
• Divisibility can often reveal interesting patterns and relationships within numbers.

Starting with the smallest number is helpful. Check each number up to the square root of the given number to see if it divides evenly.

Prime Numbers

Prime numbers play a special role in mathematics. A prime number is a number greater than 1, which can only be divided by 1 and itself without leaving any remainder. This makes them simple, yet fundamental building blocks of mathematics. For example, 11 is a prime number. It can only be divided equally by 1 and 11.

• Prime numbers cannot be formed by multiplying two smaller natural numbers.
• They have exactly two distinct positive divisors: 1 and the number itself.
• Understanding prime numbers assists in grasping more complex mathematical concepts such as factorization.

Identifying prime numbers can be done quickly by checking divisibility by each integer up to its square root.

Whole Numbers

Whole numbers are simple yet foundational in mathematics. They are numbers without fractions or decimals. These numbers include 0, 1, 2, 3, and continue infinitely. Whole numbers are straightforward because they are simply numbers we count with.

• Whole numbers include 0 and all positive numbers without fractions.
• They are used to count, order, or represent quantities.
• Understanding how whole numbers interact with each other, such as through factorization, is important for mastering basic math.

When finding factors, we are searching for whole numbers that evenly divide into our original number without leaving a fraction or remainder.