The sum of any two even numbers is always an even number.

Let's denote two even numbers as 2n2n and 2m2m, where nn and mm are integers.

The sum of these two even numbers is: 2n+2m=2(n+m)2n+2m=2(n+m)

Since nn and mm are integers, n+mn+m is also an integer. Therefore, the sum 2(n+m)2(n+m) is an even number because it is divisible by 22. Thus, the sum of any two even numbers is an even number.

To find the highest common factor (HCF) of two numbers, we find the largest number that divides both of them without leaving a remainder.

For the numbers 5 and 7, the only common factor they share is 1, since both 5 and 7 are prime numbers. Since there are no other common factors besides 1, the HCF of 5 and 7 is 1.

To find the factors of a number, you need to identify all the numbers that divide the given number evenly, including 1 and the number itself.

Let's find the factors for each:

(a) Factors of 12: 1, 2, 3, 4, 6, 12

(b) Factors of 18: 1, 2, 3, 6, 9, 18

These are all the possible factors of 12 and 18.

The only even prime number is 2. It's the only even number that is prime because by definition, an even number is divisible by 2 and no other even number (except for -2) can be prime because they are all divisible by 2 and thus not satisfying the definition of a prime number. So, the number which is even as well as prime is 2.

The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed uniquely as a product of prime numbers, except for the order of the factors. In other words:

• Every positive integer greater than 1 can be factored into a product of prime numbers.
• This factorization is unique up to the order of the factors.

For example, let's take the number 12:

12=2×2×312=2×2×3

This is one possible prime factorization of 12. According to the Fundamental Theorem of Arithmetic, any other factorization of 12 into prime numbers will involve the same prime factors (in this case, 2 and 3), just in a different order.

This theorem is fundamental in number theory and is essential for many mathematical proofs and concepts, including the understanding of factors, divisors, and properties of numbers.