31. Determine whether each of the following numbers is prime or composite.
(a) 231 (b) 393
Try the divisibility rules:
A number is divisible by 2 if the last digit is even.
(a) ..1 NO
(b) ..3 NO
A number is divisible by 3 if the sum of the digits is divisible by three.
(a) 2+3+1=6/3=2 with no remainder
(b) 3+9+3=12/3=4 with no remainder
Can you make the conclusions?
To determine whether a number is prime or composite, we need to understand the definitions of prime and composite numbers.
1. Prime number: A prime number is a positive integer greater than 1 that has no positive divisors other than 1 and itself. In other words, a prime number can only be divided evenly by 1 and itself.
2. Composite number: A composite number is a positive integer greater than 1 that has at least one positive divisor other than 1 and itself. In other words, a composite number can be divided evenly by at least one number other than 1 and itself.
To determine whether a number is prime or composite, we can follow these steps:
(a) For 231:
- To check whether 231 is divisible by any number from 2 to the square root of 231 (rounded down), we will divide 231 by each number individually.
- Starting from 2 and going up to the square root of 231 (which is approximately 15.19, so we can use integers up to 15), we divide 231 by each number and check if it leaves a remainder of 0.
- If any of these divisions give us a remainder of 0, then 231 is not a prime number and is therefore composite.
- Performing these divisions, we find that 231 is divisible evenly by 3 and 77. Thus, 231 is composite.
(b) For 393:
- Again, we will check for division by each number from 2 to the square root of 393 (which is approximately 19.82, so we can use integers up to 19).
- By performing the divisions, we determine that 393 is evenly divisible by 3, 9, 13, and 17. Thus, 393 is composite.
In conclusion, both numbers 231 and 393 are composite numbers.