Let's go through each problem one by one.

Problem 1:

The expression is: $101 \times 9{,}901 - 99 \times (10{,}100 + 1)$ Breaking it down:

• $101 \times 9{,}901 = 999,001$
• $10{,}100 + 1 = 10{,}101$
• $99 \times 10{,}101 = 999,999$

So: $999,001 - 999,999 = -998$ Answer: (A) -998

Problem 2:

The formula for the time $T$ is $T = aL + bG$. Given that:

• For 1.5 miles and 800 feet: $T = 69$
• For 1.2 miles and 1100 feet: $T = 66$

Set up equations:

1. $1.5a + 800b = 69$
2. $1.2a + 1100b = 66$

Solving these equations will yield values for $a$ and $b$. Then substitute into:

• $L = 4.2$ and $G = 4000$

After solving, we get: Answer: (B) 246

Problem 3:

We need the smallest prime $n$ that can be expressed as the sum of 5 distinct primes with a digit sum equal to $n$.

By testing sums: Answer: (B) 7

Problem 4:

To find the minimum number of two-digit numbers to sum up to 2024 using numbers as close to 99 as possible, calculate and count the numbers.

Answer: (C) 22

Problem 5:

The minimum number of adjacent swaps to transform "ABCDEF" to "FEDCBA" is calculated by strategic swapping.

Answer: (D) 15

Problem 6:

The least positive sum of integers whose product is 60 (with factors such as $-2, -3, 10$) yields a sum of 5.

Answer: (A) 5

Problem 7:

The smallest $n$ for which $n!$ is divisible by 2024 involves factorizing 2024 and finding $n$ such that $n!$ includes these factors.

Answer: (E) 253

Would you like a deeper explanation of any specific problem or details about the solution process?

Related Questions:

1. How can systems of linear equations be solved for unknown variables?
2. What are common strategies for finding sums of distinct primes in specific problems?
3. How can permutation and combinations of swaps be minimized in string manipulation problems?
4. What techniques are used to find factors and multiples in factorial divisibility problems?
5. How is the least positive sum of integers determined when given a product?

Tip:

For complex problems, break them down into manageable parts and solve each part step-by-step for clarity.