If a number is divisible by another number (in this case, 12), it must also be divisible by all the factors of that number. The factors of 12 are 1, 2, 3, 4, 6, and 12. Of the choices given, only 6 is a factor of 12. Therefore, any number divisible by 12 must also be divisible by 6. For example, 36 is divisible by 12 but not by 8 or 9. 12 is divisible by 12 but not by 24.

First, find the total number of two-digit multiples of 4. The smallest is 12 (\(4 \times 3\)) and the largest is 96 (\(4 \times 24\)). There are \(24 - 3 + 1 = 22\) such numbers. Next, find the number of two-digit multiples of 8. The smallest is 16 (\(8 \times 2\)) and the largest is 96 (\(8 \times 12\)). There are \(12 - 2 + 1 = 11\) such numbers. Every multiple of 8 is also a multiple of 4. The question asks for the numbers that are multiples of 4 but NOT multiples of 8. So, we subtract the count of multiples of 8 from the count of multiples of 4: \(22 - 11 = 11\).

First, list the factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. From this list, identify the multiples of 4: 4, 12, 20, 60. From this new list, select numbers greater than 5: 12, 20, 60. Finally, from this list, find the number that is not divisible by 5. Both 20 and 60 are divisible by 5, so the only remaining number is 12.

To check if a number is divisible by 4, we check if the number formed by its last two digits is divisible by 4. For 1998, 98 is not divisible by 4. For 2002, 02 (or 2) is not divisible by 4. For 2010, 10 is not divisible by 4. For 2016, 16 is divisible by 4 (\(16 \div 4 = 4\)), so 2016 was a leap year.

First, list the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36. Next, find a pair of different factors from this list that add up to 21. By checking pairs, we find that \(3 + 18 = 21\). The two factors are 3 and 18. Finally, the question asks for the product of these two factors, which is \(3 \times 18 = 54\).

First, list the factors of 24: 1, 2, 3, 4, 6, 8, 12, 24. Second, check which of these factors is also a multiple of 3. The multiples of 3 are 3, 6, 9, 12, 15, etc. The numbers that are in both lists are 3, 6, 12, and 24. Of the answer choices provided, only 12 fits both descriptions. 8 is a factor but not a multiple of 3. 9 is a multiple of 3 but not a factor of 24. 48 is a multiple of both 3 and 24, but it is not a factor of 24.

For a number to be divisible by 9, the sum of its digits must be a multiple of 9. The sum of the known digits is \(5 + 8 + 2 = 15\). The next multiple of 9 after 15 is 18. To make the sum of the digits equal 18, the missing digit must be \(18 - 15 = 3\). So the number is 5,832.

This problem is asking for the greatest common factor (GCF) of 32 and 40. The factors of 32 are 1, 2, 4, 8, 16, 32. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, 40. The common factors are 1, 2, 4, and 8. The greatest of these is 8. Therefore, the teacher can make 8 identical kits, each with 4 pencils (32/8) and 5 erasers (40/8).

First, find the prime factors of 42. We can start by dividing by the smallest prime number, 2: \(42 = 2 \times 21\). Then, find the factors of 21: \(21 = 3 \times 7\). Both 3 and 7 are prime numbers. So, \(42 = 2 \times 3 \times 7\). These are three different prime numbers. The sum of these prime numbers is \(2 + 3 + 7 = 12\).

A number with exactly three factors is the square of a prime number. For example, the factors of 9 (which is \(3^2\)) are 1, 3, and 9. The factors of 25 (which is \(5^2\)) are 1, 5, and 25. Prime numbers have exactly two factors. Not all numbers with three factors are odd (e.g., 4 has factors 1, 2, 4). The square of a composite number has more than three factors (e.g., 16, which is \(4^2\), has factors 1, 2, 4, 8, 16).