Problem 24 - Fundamentals of Mathematics

Question

Write all the divisors of 18 .

Step-by-Step Solution

Answer

The divisors of 18 are 1, 2, 3, 6, 9, and 18.

Step 1: Understand the Problem

To find the divisors of a number, we need to determine all the whole numbers that can divide the given number, 18, without leaving a remainder.

Step 2: Check from 1 to 18

We will check each number from 1 to 18 to see if it divides 18 without leaving a remainder. A divisor of 18 will satisfy the condition that when 18 is divided by it, the remainder is zero.

Step 3: Identify Divisors

Start with 1. \(18 \div 1 = 18\) so 1 is a divisor. Next, use 2. \(18 \div 2 = 9\), which is a whole number, so 2 is a divisor. Continue this process for each number up to 18.

Step 4: List the Divisors

After checking each number from 1 to 18, we find that the divisors of 18 are 1, 2, 3, 6, 9, and 18. These numbers evenly divide 18 without leaving any remainder.

Key Concepts

Whole Numbers

Whole numbers are the set of numbers that include zero and all the positive numbers without any fractions or decimals. They are 0, 1, 2, 3, and so on. Whole numbers are crucial in identifying divisors because they are the numbers we use when checking for exact divisibility.

When we say a number is a divisor of another, we mean that dividing the two results in a whole number. For example, when we divide 18 by one of its divisors like 3, the result is 6, which is a whole number. Therefore, 3 is a divisor of 18.

Whole numbers are an important part of basic mathematics, as they form the building blocks for more complex mathematical concepts. Understanding them is essential for solving division problems and finding divisors.

Remainder

The remainder is what is left over after a division problem is completed when the numbers do not divide evenly. If there is no remainder, it means the division resulted in a whole number. This concept is key when we are looking for divisors of a number.

For example, if we divide 18 by 5, we do not get a whole number. Instead, we get a quotient of 3 with a remainder of 3, since 3 times 5 is 15 and 18 minus 15 leaves 3.

• A remainder of zero indicates a perfect division.
• If the remainder is non-zero, then the divisor does not divide the number fully.

In the exercise of finding all divisors of 18, we ensure that all the chosen divisors leave a remainder of zero. This confirms that they divide the number completely.

Division

Division is the process of determining how many times a number, known as the divisor, can be contained within another number, known as the dividend. It is one of the four fundamental arithmetic operations. To ascertain the divisors of a given number like 18, we engage in division until we find all numbers that divide 18 exactly, yielding a whole number as a quotient without a remainder.

In our specific case of finding divisors, it involves dividing 18 by each whole number from 1 up to 18.

• If the division results in a whole number with no remainder, the divisor is recorded as a legitimate divisor of 18.
• If there is a remainder, that particular number is not a divisor of 18.

This process helps list the complete set of divisors of a number, allowing us to fully understand the factors contributing to its make-up in multiplication scenarios.