Here’s the solution for each of the questions visible in the uploaded image:

Question 1:

The prime factorization of 2100 is:

Break down 2100:
$2100 = 2 \times 2 \times 3 \times 5 \times 5 \times 7$

Answer: (c) $2 \times 2 \times 3 \times 5 \times 5 \times 7$

Question 2:

The smallest 4-digit number when expressed as a product of primes is:

The smallest 4-digit number is 1000. Its prime factorization:
$1000 = 2 \times 2 \times 2 \times 5 \times 5 \times 5$

Answer: (a) $2 \times 2 \times 2 \times 5 \times 5 \times 5$

Question 3:

The prime factorization of 640 is:

Break down 640 into prime factors:
$640 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 2^{10}.$

Answer: (a) $2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2$

Question 4:

A common multiple of both 9 and 7 is A. This number is between 1200 and 1300.

The least common multiple (LCM) of $9$ and $7$ is: $\text{LCM}(9, 7) = 63$ Now, find multiples of 63 between 1200 and 1300: $63 \times 19 = 1197 \quad (\text{too small}), \quad 63 \times 20 = 1260 \quad (\text{fits range}).$

Answer: (c) 1260

Question 5:

The sum of the first four multiples of 6 is:

The first four multiples of $6$ are:
$6, 12, 18, 24$
Sum: $6 + 12 + 18 + 24 = 60$

Answer: (a) 66

Question 6:

The sum of the first five multiples of 6 is:

The first five multiples of $6$ are:
$6, 12, 18, 24, 30$
Sum: $6 + 12 + 18 + 24 + 30 = 90$

Answer: (a) 90

Question 7:

The difference between the tens digit and the units digit of the sum of the first five multiples of 6 is:

From Question 6, the sum of the first five multiples of $6$ is: $90$
The tens digit is $9$, and the units digit is $0$. Difference: $9 - 0 = 9$

Answer: (d) 9

Question 8:

Which of the following is NOT a factor of 316?

The prime factorization of $316$ is: $316 = 2 \times 2 \times 79$
The factors are: $1, 2, 4, 79, 158, 316$
Among the options, $8$ is NOT a factor.

Answer: (b) 8

Question 9:

What is the prime factorization of 37800?

Break down $37800$:
$37800 = 2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 7$

Answer: (c) $2 \times 2 \times 3 \times 3 \times 3 \times 5 \times 5 \times 7$

Question 10:

Factors of 30 are:

The factors of $30$ are: $1, 2, 3, 5, 6, 10, 15, 30$

Answer: (b) $1, 2, 3, 5, 6, 10, 15, 30$

Would you like detailed explanations for any of these, or should I proceed to additional problems? Let me know!

Related Questions:

1. How do you find the least common multiple (LCM) of two numbers?
2. What is the difference between factors and multiples?
3. How can you verify the prime factorization of a number?
4. Why is prime factorization useful in solving problems?
5. How do you find the sum of consecutive multiples efficiently?

Tip:

To quickly calculate sums of consecutive multiples, use the formula for the sum of an arithmetic sequence: $S = \frac{n}{2} \times (\text{first term} + \text{last term}),$ where $n$ is the number of terms.