Answer:
10. Given a, b E Z, if a²(b² - 2b) is odd, then both a and b must be odd numbers. This is because the only way for the product of two numbers to be odd is if both of the numbers are odd. In other words, if a and b are both even, then a²(b² - 2b) would be an even number, and if one of a and b is odd and the other is even, then a²(b² - 2b) would be an even number. So the only way for a²(b² - 2b) to be odd is if both a and b are odd.
11. Given m, n E Z, if 25 + mn is divisible by 5, then either m or n must be divisible by 5. This is because if both m and n are not divisible by 5, then the product mn will not be divisible by 5, and 25 + mn would not be divisible by 5. However, if either m or n is divisible by 5, then the product mn will be divisible by 5, and 25 + mn would be divisible by 5. So if 25 + mn is divisible by 5, then either m or n must be divisible by 5.
12. If m, n E Z and m + n is even, then m² + n² is also even. This is because the sum of two even numbers is always even, and the square of an even number is always even. So if m + n is even, then both m and n must be even, and m² and n² would both be even. Therefore, the sum m² + n² is also even.
13. An integer n is even if and only if n² + 2n + 9 is odd. This is because the square of an even number is always even, and the sum of two even numbers is always even. So if n is even, then n² and 2n are both even, and the sum n² + 2n + 9 would be odd. On the other hand, if n² + 2n + 9 is odd, then the sum of the first two terms, n² + 2n, must be even. Since the square of an even number is always even, this means that n must be even. So n is even if and only if n² + 2n + 9 is odd.
14. If a is an odd integer, then a² + 3a + 5 is also odd. This is because the product of two odd numbers is always odd, and the sum of two odd numbers is always even. So if a is odd, then a² and 3a are both odd, and the sum a² + 3a + 5 would be even. Since the sum of two odd numbers is always even, this means that a² + 3a + 5 is odd. Therefore, if a is an odd integer, then a² + 3a + 5 is also odd.
Explanation: