# Show that 500 is not a perfect square

The method to represent and work with numbers is known as the number system. A number system is a system of writing to represent numbers. It is the mathematical notation used to represent numbers of a given set by using digits or other symbols. It allows us to operate arithmetic operations such as division, multiplication, addition, subtraction.

Some important number systems are as follows:

- Decimal Number System
- Binary Number System
- Octal Number System
- Hexadecimal Number System

Let’s see about all these number systems in detail.

**Decimal Number System**

The decimal number system consists of ten digits i.e. from 0 to 9. The base of the decimal number system is 10. These digits can be used to represent or express any numeric value.

For example, the decimal number 245 consists of the digit 5 in ones place, the digit 4 in tens place and the digit 2 in hundreds place which can be represented as :

(2×10^{2}) + (4×10^{1}) + (5×10^{0})

= (2×100) + (4×10) + (5) { where, 10^{0} = 1}

= 200 + 40 + 5

= 245

**Binary Number System**

The binary number system consists of only two digits i.e. 0 and 1. The base of the binary number system is 2. The digital computer represents all kinds of data in a binary number system.

For example, convert 100111_{2} into decimal number system.

(100111)_{2} = 1×2^{5} + 0×2^{4} + 0×2^{3} + 1×2^{2} + 1×2^{1} + 1×2^{0}

= 32 + 0 + 0 + 4 + 2 + 1

= (39)_{10}

**Octal Number System**

The octal number system consists of digits from 0 to 7. The base of the octal number system is 8. Octal number systems are basically used in computer applications.

For example, convert 545_{8} into decimal.

545_{8} = 5×8^{2} + 4×8^{1} + 5×8^{0}

= 320 + 32 + 5

= 357_{10}

**Hexadeciamal Number System**

In the hexadecimal number system, numbers are first represented from digits 0 to 9 as decimal number system, and then the numbers are represented using alphabets from A to F. The base of the hexadecimal number system is 16.

For example, convert 15DB_{16} to decimal.

15DB_{16} = 1×16^{3} + 5×16^{2} + 13×16^{1} + 11×16^{0}

= 4096 + 1280 + 208 + 11

= 5595_{10}

### What is a Perfect Square?

A perfect square is a number that can be represented as the product of two equal integers.

For example, 36 is a perfect square as it has two equal integers i.e. (6×6 = 36), whereas 21 is not a perfect square because it does not have two equal integers i.e. (7×3 = 21).

### Show that 500 is not a perfect square

In the above-given question first, we have to find the factors of 500.

So, the factors of 500 are 2 × 2 × 5 × 5 × 5.

In the factors of 500, we have the pairs (2×2) and (5×5) but one 5 is left alone. Therefore, 500 is not a perfect square.

### Similar Questions

**Question 1: Is 441 a perfect square?**

**Answer:**

In the above-given question first, we have to find the factors of 441.

So, the factors of 441 are 3 × 7 × 3 × 7.

In the factors of 441, we have the pairs (3×3) and (7×7) and there is no single digit left alone. Therefore, 441 is a perfect square.

**Question 2: Is 600 a perfect square?**

**Answer:**

In the above-given question first, we have to find the factors of 600.

So, the factors of 600 are 2 × 2 × 2 × 3 × 5 × 5.

In the factors of 600, we have pairs (2×2) and (5×5) but 2 and 3 are left alone. Therefore, 600 is not a perfect square.

**Question 3: Is 324 a perfect square?**

**Answer:**

In the above-given question first, we have to find the factors of 324.

So, the factors of 324 are 2 × 2 × 3 × 3 × 3 × 3.

In the factors of 324, we have pairs (2×2), (3×3), and (3×3) and there is no single digit left alone. Therefore, 324 is a perfect square.