Find two different PRIME numbers whose sum is a composite number. Explain the DIVISIBILITY rule that proves the sum is composite.
One example of two different prime numbers whose sum is a composite number is 2 and 5. The sum of 2 and 5 is 7, which is a prime number.
To explain the divisibility rule that proves the sum is composite, we need to understand the concept of divisibility. A number is said to be divisible by another number if it can be divided evenly without leaving a remainder.
When we add two prime numbers, the resulting sum can only be even if at least one of the prime numbers is 2. This is because any odd number added to another odd number always results in an even number. However, the sum of two odd primes will always be an even number since odd + odd = even.
Since 2 is the only even prime number, any other prime number added to 2 will result in an even sum. An even number is always composite because it is divisible evenly by 1, itself, and at least one other number (2).
In the example of 2 and 5, the sum is 7, which is not divisible evenly by any number other than 1 and itself. Hence, the sum is a prime number.
Therefore, to find two different prime numbers whose sum is a composite number, we need to select two odd prime numbers excluding 2.
To find two different prime numbers whose sum is a composite number, we can consider the prime numbers 2 and 5.
First, let's determine their sum:
2 + 5 = 7.
In this case, the sum of the prime numbers (2 + 5 = 7) is not composite. However, we can still explain a divisibility rule that proves the sum of two prime numbers could be composite.
The divisibility rule that we can consider is the fact that any prime number (except 2) is an odd number. Adding two odd numbers, such as two different prime numbers, will always result in an even number.
When we add 2 and 5, the sum is 7, which is not divisible by any other number except for 1 and itself. Thus, in this example, the sum is not composite. However, if we were to add two larger odd prime numbers, their sum could be divisible by 2, making it composite.
To find two different prime numbers whose sum is a composite number, we need to understand the concept of divisibility and apply it. The divisibility rule we will use is the fact that if a number is composite, it is divisible by at least one prime number other than itself.
To begin, let's list the prime numbers:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, ...
Now, we need to find two prime numbers whose sum is a composite number. Let's try an example:
Let's take the prime numbers 2 and 7. The sum of these two primes is 2 + 7 = 9. To check if 9 is a composite number, we can look for a prime number that divides evenly into 9, other than 9 itself.
Using the list of prime numbers, we see that 3 is a prime number that divides evenly into 9. Therefore, the sum of the primes 2 and 7 is a composite number (9), as it is divisible by a prime number other than itself.
So, the pair of prime numbers (2, 7) satisfies the condition of the question.
In summary, the divisibility rule that proves the sum of two prime numbers is composite is that if the sum is composite, it will be divisible by at least one prime number other than itself. By checking for divisibility using prime numbers, we can determine if a sum is composite or not.