How Many Factors of 1080 are Perfect Squares

Introduction

The study of quality, shape, structure etc is called Maths. Maths deals with learning the logic behind a particular concept and then implementing it to solve daily real-life situations. Mathematics helps students to grow their problem-solving and logic-building skills. Generally, Maths includes four types of operators- addition, subtraction, multiplication, and division. In mathematics, we also deal with Square and Square roots and Cube and Cube Roots of a number. Let's learn Square and square roots in detail.

Perfect Square of a Number

Perfect Square numbers are numbers found by multiplying the number by itself. In other words, we can also say if a natural number is multiplied by itself then the resulting number is called the squared number or the square of a given number or a simply perfect square. The square of a number always ends with either 0, 4, 5, 6, or 9. Square numbers can never be negative because if a negative number is multiplied by the same negative number then it results in a positive number. Hence square number is always a positive number.

Let's take some examples of calculating the Square of a number:

Example 1: Find the square of 25.

Ans:

\begin{equation}

25 * 25=625

\end{equation}

Hence 625 is the square of 25.

Example 2: Find the square of 13.

Ans:

\begin{equation}

13 * 13=169

\end{equation}

Hence 169 is the square of 13.

Example 3: Find the square of 15.

Ans:

\begin{equation}

15 * 15=225

\end{equation}

Hence 225 is the square of 15.

Square Root of a Number

It is just the inverse operation of the squaring of a number. The square root of a number is the number that when multiplied by itself gives the original number. In other words, it is the value of power ½ of that number. The square root is represented using the symbol '√ '. This symbol is called a radical whereas the number under the square root symbol is called the radicand. If the square root of the number gives the whole number as a result then that number is called a perfect square.

Let’s take some examples to calculate the square root of a number.

Example 1: Find the square root of 169.

Ans:

\begin{equation}

\begin{aligned}

& 13 * 13=169 \\

& \sqrt{169}=13

\end{aligned}

\end{equation}

Hence 13 is the square root of 169.

Example 2: Find the square root of 144.

Ans:

\begin{equation}

\begin{aligned}

& 12 * 12=144 \\

& \sqrt{144}=12

\end{aligned}

\end{equation}

Hence 12 is the square root of 144.

Prime Numbers

The numbers which are divisible only by 1 and the number itself are called Prime Numbers. 1 is neither a prime number nor a composite number. The smallest prime number is 2 which has only two factors 1 and 2

Let’s take some examples:

Example 1: Check whether 29 is a prime number or not.

Ans: Factors of 29 are 1, 29. Since it has two factors only that is 1 and the number itself, Hence it is a prime number.

Example 2: Check whether 53 is a prime number or not.

Ans: Factors of 53 are 1, 53. Since it has two factors only that is 1 and the number itself, Hence it is a prime number.

Example 3: Check whether 48 is a prime number or not.

Ans: Factors of 53 are 1, 2, 3,4, 6, 8, 12 and 48. Since it has more than two factors so it is not a prime number.

Composite Numbers

The numbers which have more than two factors are called Composite Numbers. In other words, we can say the numbers which are not prime are composite numbers. There is a fact related to Composite Numbers that Every even number greater than 2 is a composite number. The smallest composite number is 4.

Let’s take some examples of Composite Numbers:

Example 1: Check whether 56 is a prime number or not.

Ans: Factors of 53 are 1, 2, 4,7,8,14,28 and 56. Since it has more than two factors, Hence it is a Composite Number.

Example 2: Check whether 66 is a prime number or not.

Ans: Factors of 66 are 1, 2, 3, 6,11, 22 and 33. Since it has more than two factors, Hence it is a Composite Number.

Example 3: Check whether 61 is a prime number or not.

Ans: Factors of 61 are 1, 61. Since it has only two factors, Hence it is not a Composite Number but a Prime Number.

Factors of 1080

\begin{equation}

\begin{aligned}

& 1 * 1080=1080 \\

& 2 * 540=1080 \\

& 3 * 360=1080 \\

& 4 * 270=1080 \\

& 5 * 216=1080 \\

& 6 * 180=1080 \\

& 8 * 135=1080 \\

& 9 * 120=1080 \\

& 10 * 108=1080 \\

& 12 * 90=1080 \\

& 15 * 72=1080 \\

& 18 * 60=1080 \\

& 208 * 54=1080 \\

& 24 * 45=1080 \\

& 27 * 40=1080 \\

& 30 * 36=1080

\end{aligned}

\end{equation}

Hence the factors of 1080 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540, 1080.

What are Perfect Squares from factors from 1080

Perfect Square numbers are the numbers which when taken a square root give the original number formed by multiplying.

Therefore the perfect squares from the factors of 1080 are 1, 4, 9 and 36.

\begin{equation}

\begin{aligned}

& 1 * 1=1 \\

& 2 * 2=4 \\

& 3 * 3=9 \\

& 6 * 6=36

\end{aligned}

\end{equation}

Hence there are four perfect squares in the factors of 1080.

Conclusion

Following points we had learnt:

• When a number is multiplied by itself then the result obtained is a perfect square.

• The perfect square of a number always ends with either 0, 4, 5, 6, or 9.

• Square root is the inverse operation of squaring a number.

• The numbers which are divisible only by 1 and the number itself are called Prime Numbers.

• Composite Numbers have more than two factors.

• Factors of 1080 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 27, 30, 36, 40, 45, 54, 60, 72, 90, 108, 120, 135, 180, 216, 270, 360, 540, 1080.

• Perfect Squares from the factors of 1080 are 1, 4, 9 and 36.