360 = 2³×3²×5¹. Number of divisors = (3+1)(2+1)(1+1) = 4×3×2 = 24.

A number is divisible by 8 if its last 3 digits form a number divisible by 8. 456 ÷ 8 = 57. So 2456 is divisible by 8.

1071 = 462×2 + 147; 462 = 147×3 + 21; 147 = 21×7 + 0. Therefore HCF = 21.

Even numbers: 2, 4, 6, ..., 100. This is AP with a=2, l=100, d=2. n = 50 terms. Sum = (n/2)(a+l) = (50/2)(2+100) = 25×102 = 2550.

To find the value of this expression, we'll factor out the smallest power of 2 and simplify using algebraic factoring.

Step 1: Factor out \(2^7\) from all terms

Each term contains at least \(2^7\) as a factor, so we can write:

Step 2: Simplify the expression in parentheses

Calculate each power of 2:

Step 3: Multiply by the factored term

Answer: \(2^{10} - 2^9 - 2^8 - 2^7 = 128\) (Option B)

The digital root is found by repeatedly summing the digits of a number until a single digit remains.

Step 1: Sum all digits of 9875

Add each digit:

Step 2: Sum the digits of the result

Since 29 is not a single digit, sum its digits:

Step 3: Sum again until single digit

Since 11 is still two digits, sum once more:

Step 4: Verify the digital root

We have reached a single digit. The digital root of 9875 is \(2\).

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Answer: The digital root of 9875 is \(2\) (Option A)

Let consecutive odd numbers be (2n-1) and (2n+1). (2n-1)(2n+1) = 323. 4n² - 1 = 323, so 4n² = 324, n² = 81, n = 9. Numbers are 17 and 19.

Let number = 5a + 2 = 7b + 3. From 5a + 2 = 7b + 3, we get 5a = 7b + 1. Testing b = 2: 7(2) + 1 = 15, a = 3. Number = 5(3) + 2 = 17. Check: 17 ÷ 5 = remainder 2, 17 ÷ 7 = remainder 3. ✓

Let number = 10a + b. (10a + b) - (10b + a) = 45. So 9a - 9b = 45, a - b = 5. Also a + b = 9. Solving: a = 7, b = 2. Number = 72.

144 = 2⁴ × 3², 108 = 2² × 3³. HCF = 2² × 3² = 4 × 9 = 36.