Numbers - Session 2

Have you ever wondered what happens when you multiply a number by itself? What do you think the result would be?Square numbers are numbers that you get when you multiply a number by itself. This means if you take any whole number and multiply it by the same number, the result is a square number. For example, if you take the number two and multiply it by two, the result is four. Similarly, if you multiply three by three, the result is nine. This pattern continues with other numbers.

A square number can be thought of as the area of a square. If you have a square with a side length of four, the area of that square would be sixteen because four multiplied by four equals sixteen. This idea helps us understand why these numbers are called square numbers.

Another important concept is square roots. A square root is the opposite of a square number. For example, the square root of sixteen is four because four multiplied by four gives you sixteen. The square root helps us find out what number was multiplied by itself to get a square number. This is an important idea in mathematics and is used when solving problems involving squares or areas.

Cube numbers are numbers that you get when you multiply a number by itself three times. For example, if you take the number two and multiply it by itself three times, you get eight. This is because two multiplied by two is four, and then multiplying four by two again gives you eight.Similarly, if you take three and multiply it by itself three times, you get twenty seven. So, cube numbers are the result of multiplying a number by itself three times.

Similar to square numbers, cube numbers can be understood by thinking about shapes. Imagine a cube, the three dimensional shape. If each side of the cube is three units long, the volume of the cube would be the result of multiplying three by three by three. This gives us the number twenty seven, which is the volume of the cube in cubic units. This is why we call them cube numbers because they represent the volume of a cube, where all the sides are equal.

Another interesting concept related to cube numbers is cube roots. Similar to square roots are the opposite of square numbers, cube roots are the opposite of cube numbers. The cube root of eight is two because two multiplied by itself three times equals eight. Cube roots are useful when you need to find the original number that was cubed.

Factors are simply numbers that divide evenly into a given number. To clarify, imagine you have a number, say twelve. Factors of twelve are all the numbers that you can multiply together to get twelve. These factors include one, two, three, four, six, and twelve itself.

This means if you take one and multiply it by twelve, or two and multiply it by six, or three and multiply it by four, you will always get twelve. Essentially, factors help break-down a number into smaller, manageable components. They are the basic building-blocks of any number.

To further explain, think of factors in terms of dividing a group of objects. Let us say you have twelve apples, and you want to organize them into smaller groups. The number one is always a factor of any number because you can divide anything by one. Similarly, twelve apples can be divided into six groups of two apples each, or four groups of three apples each, or two groups of six apples each or three groups of four apples each. Factors help us understand the structure of a number by showing us what numbers can divide it evenly. It’s like finding out who can create groups of the same size out of it.

Now, common factors are those factors that two or more numbers share. In simpler words, if you are comparing two numbers, common factors are the numbers that divide evenly into both of those numbers. Let us take the example of twelve and eighteen. To find the common factors, you first list the factors of twelve, which are one, two, three, four, six, and twelve. Then, you list the factors of eighteen, which are one, two, three, six, nine, and eighteen. The common factors between twelve and eighteen are one, two, three, and six. These numbers are factors of both twelve and eighteen.

Now you are given the numbers twenty and thirty. Can you find their common factors? The first thing you need to do is find the factors of each number. For the number twenty, you start by looking at all the numbers that can divide twenty evenly, meaning without leaving a remainder. These factors are one, two, four, five, ten, and twenty. These are all the numbers that, when multiplied in pairs, give twenty. For example, one times twenty equals twenty, two times ten equals twenty, and so on.

Next, let us do the same for the number thirty. The factors of thirty are one, two, three, five, six, ten, fifteen, and thirty. These numbers can divide thirty evenly. For example, one times thirty equals thirty, two times fifteen equals thirty, and so on.Now, to find the common factors, you compare the two lists of factors. You look for numbers that appear in both lists. If you compare the factors of twenty and thirty, you will find that the numbers one, two, five, and ten appear in both lists. These are the common factors of twenty and thirty.

The highest common factor is the largest number that can divide two or more numbers exactly without leaving a remainder. It is a way of finding the greatest shared factor between two numbers. For example, if you are given the numbers twenty four and thirty six, you would start by identifying all the factors of each number.

To find the factors of twenty four, you would start by listing all the numbers that can divide twenty four evenly. These are one, two, three, four, six, eight, twelve, and twenty four. These are the factors of twenty four because you can multiply them by other numbers to get twenty four. For example, one multiplied by twenty four equals twenty four, and two multiplied by twelve equals twenty four.

Next, you do the same thing for thirty six. The factors of thirty six are one, two, three, four, six, nine, twelve, eighteen, and thirty six. These are all the numbers that divide thirty six evenly.Now, to find the highest common factor, you look at both sets of factors and pick out the numbers that are the same in both sets. In this case, the numbers that are common to both lists are one, two, three, six, and twelve. The largest of these is twelve. So, the highest common factor of twenty four and thirty six is twelve. Can you tell the highest common factor of twenty five and fifty?