Identify the number 113 as either prime or composite. Show or explain why

does it have any prime factors?

Here is a neat exercise...what are the factors of 111? (note the sum of digits is three)

So because it doesn't have any prime factors, it is a prime factor itself?

To determine whether the number 113 is prime or composite, we can apply a method called trial division. Here's how it works:

1. Start by dividing 113 by the smallest prime number, which is 2. We'll check if 113 is divisible evenly by 2.
- 113 ÷ 2 = 56 remainder 1

2. Next, divide 113 by the next prime number, which is 3.
- 113 ÷ 3 = 37 remainder 2

3. Continue dividing 113 by prime numbers, incrementing each time, until you reach the square root of 113 or a divisor greater than that.
- In this case, we need to check division by 5 and 7.

4. Divide 113 by 5.
- 113 ÷ 5 = 22 remainder 3

5. Finally, divide 113 by 7.
- 113 ÷ 7 = 16 remainder 1

Now, let's evaluate the results. If any of the divisions resulted in an even quotient (no remainder), the number 113 would be composite. However, since none of the divisions gave an even quotient, we know that 113 is not divisible by any prime numbers between 2 and 10 (inclusive), except for 113 and 1.

Therefore, the number 113 is a prime number as it is divisible only by 1 and itself.