# What is the smallest 3 digit prime number?

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, and 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 153 consists of the digit 3 in ones place, the digit 5 in the tens place, and the digit 1 in hundreds place which can be represented as,

(1 × 10^{2}) + (5 × 10^{1}) + (3 × 100^{0 })

= (1 × 100) + (5 × 10) + (3 × 1) { where, 100 = 1}

= 100 + 50 + 3

= 153

### Prime Number

In the number system, Prime Numbers are those numbers that have only two factors that is 1 and the number itself. In other words, the prime numbers are those numbers that are exactly divisible by 1 and the number itself.

- A Prime number should contain exactly two factors.
- A prime number should be divisible 1 and the number itself.

Let us assume p is a prime number then p has only 2 factors that are 1 and p itself. Any number which does not follow this is termed a composite number. For example factors of 8 are 1, 2, 4, and 8, which are four factors in total. But factors of 5 are 1 and 5 itself, totally two factors. Hence, 5 is a prime number but 8 is not a prime no, instead, it is a composite number. First Ten Natural Prime Numbers are – 2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Below is the list of prime numbers from 1 to 100,

**Note **1 is a non-prime number because according to the definition, a prime number should contain only two factors but 1 has only one factor. Therefore 1 is not a prime number.

**List of Prime Number between 1 to 100 **

The prime numbers are the numbers that have only two factors which are 1 and the number itself. Let’s see some prime numbers that are present between 1 and 100, along with their factors.

2 = (1, 2), 3 = (1, 3), 5 = (1, 5), 7 = (1, 7), 11 = (1, 11), 13 = (1, 13), 17 = (1, 17), 19 = (1, 19), 23 = (1, 23), 29 = (1, 29), 31 = (1, 31), 37 = (1, 37), 41 = (1, 41), 43 = (1, 43), 53 = (1, 53), 59 = (1, 59), 61 = (1, 61), 67 = (1, 67), 71 = (1, 71), 73 = (1, 73), 79 = (1, 79), 83 = (1, 83), 89 = (1, 89), 97 = (1, 97).

### What is the smallest 3 digit prime number?

The smallest 3 digit number in the number system is 100 because if we subtract -1 from the number (100) it becomes 99 which is a 2 digit number. So, 100 is the smallest 3 digit number in the number system. Factors of 100 are = 1, 2, 4, 5, 10, 20, 25, 50, and 100.

A prime number has only two factors i.e. it is divisible by either 1 or by the number itself. 100 doesn’t satisfy all the conditions of a prime number that’s why100 is not a prime number. Let’s increase the value (100) by +1 then it becomes 101. Factors of 101 are = 1 and 101. A prime number has only two factors i.e. it is divisible by either 1 or by the number itself. 101 satisfy all the conditions of a prime number. So, 101 is a prime number.

**Therefore, 101 is the smallest 3-digit prime number in the number system.**

### Sample Problems

**Question 1: What is the largest 3 digit prime number?**

**Solution :**

First, find the largest 3 digit number in the number system. The largest 3 digit number in the number system is 999 because if +1 is added to the number (999) it becomes 1000 which is a 4 digit number. So 999 is the largest 3 digit number in the number system.

Now check all the factors of the numbers in the reverse format and stop when the number has only 2 factors.

- Factors of 999 are = 1, 3, 9, 27, 37, 111, 333, 999.
- Factors of 998 are = 1, 2, 499, 998
- Factors of 997 are = 1, 997
A prime number has only two factors i.e. it is divisible by either 1 or by the number itself. As 999 has 8 factors and it doesn’t satisfy all the conditions of a prime number that’s why 999 is not a prime number.

Similarly, 998 has more than 2 factors and it doesn’t satisfy all the conditions of a prime number. So 998 is not a prime number, but 997 has only 2 factors that satisfy all the conditions of a prime number. So 997 is a prime number.

Therefore, 997 is the largest 3-digit prime number in the number system.

**Question 2: What is the smallest 2 digit prime number?**

**Solution: **

The smallest 2 digit number in the number system is 10 because if we subtract -1 from the number (10) it becomes 9 which is a 1 digit number. So 10 is the smallest 2 digit number in the number system. Factors of 10 are = 1, 2, 5, 10.

A prime number has only two factors i.e. it is divisible by either 1 or by the number itself. 10 doesn’t satisfy all the conditions of a prime number that’s why10 is not a prime number. Let’s increase the value (10) by +1 then it becomes 11. Factors of 11 are = 1 and 11

A prime number has only two factors i.e. it is divisible by either 1 or by the number itself. 101 satisfy all the conditions of a prime number. So, 11 is a prime number.

Therefore, 11 is the smallest 2-digit prime number in the number system.

**Question 3: What is the largest 2 digit prime number?**

**Solution : **

First, find the largest 2 digit number in the number system. The largest 2 digit number in the number system is 99 because if +1 is added to the number (99) it becomes 100 which is a 3 digit number. So 99 is the largest 2 digit number in the number system.

Now check all the factors of the numbers in the reverse format and stop when the number has only 2 factors.

- Factors of 99 are = 1, 3, 9, 11, 33, and 99.
- Factors of 98 are = 1, 2, 7, 14, 49, 98.
- Factors of 97 are = 1, 97
It is known that a prime number has only two factors i.e. it is divisible by either 1 or by the number itself, 99 has 6 factors and it doesn’t satisfy all the conditions of a prime number that’s why 99 is not a prime number. Similarly, 98 has more than 2 factors and it doesn’t satisfy all the conditions of a prime number. So 98 is not a prime number, but 97 has only 2 factors that satisfy all the conditions of a prime number. So 97 is a prime number.

Therefore, 97 is the largest 2-digit prime number in the number system.