1. Problem: Explore rules for sums, products, and differences involving even and odd numbers. 2. Rule (i): Sum of two even numbers. - Let even numbers be $2a$ and $2b$, where $a,b$ are integers. - Their sum: $$2a + 2b = 2(a+b)$$ - Since $a+b$ is an integer, sum is even. 3. Rule (ii): Product of two even numbers. - Using $2a$ and $2b$, product: $$ (2a)(2b) = 4ab = 2(2ab) $$ - Since $2ab$ is integer, product is even. 4. Rule (iii): Sum of an even and an odd number. - Even number: $2a$, odd number: $2b+1$. - Sum: $$2a + (2b+1) = 2(a+b) + 1$$ - This is odd since it is $2 imes$ integer + 1. 5. Rule (iv): Product of an even and an odd number. - Even: $2a$, odd: $2b+1$. - Product: $$ (2a)(2b+1) = 2a(2b+1) = 2(a(2b+1)) $$ - Even since it is 2 times an integer. 6. Rule (v): Difference of two odd numbers. - Odd numbers: $2a+1$, $2b+1$. - Difference: $$ (2a+1) - (2b+1) = 2a - 2b = 2(a-b) $$ - Even because $a-b$ is integer. 7. Rule (vi): Difference of two even numbers. - Even numbers: $2a$, $2b$. - Difference: $$ 2a - 2b = 2(a-b) $$ - Even because $a-b$ is integer. Final summary: - Sum of two evens is even. - Product of two evens is even. - Sum of even and odd is odd. - Product of even and odd is even. - Difference of two odds is even. - Difference of two evens is even.