This is a set of math problems covering Roman numerals, subtraction, prime factorization, HCF, and LCM.

Question

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Here are the solutions to the questions:

1. Express DXIV in Arabic numerals.

Break down the Roman numeral into its components.
D = 500
X = 10
IV = 4 (since I (1) comes before V (5), it means $5 - 1 = 4$ )
Add the values.
$500 + 10 + 4 = 514$
The final answer is $\boxed{\text{b) 514}}$

2. Subtract 1543 from 2993.
Set up the subtraction.
We need to calculate $2993 - 1543$ .
Perform the subtraction.
$2993 - 1543 = 1450$
The final answer is $\boxed{\text{a) 1450}}$

3. Which of the following represents the prime factors of 30?
Find the prime factors of 30.
A prime factor is a prime number that divides the given number exactly.
Divide 30 by the smallest prime number, 2:
$30 \div 2 = 15$
Divide 15 by the smallest prime number that divides it, which is 3:
$15 \div 3 = 5$
5 is a prime number.
List the prime factors.
The prime factors of 30 are 2, 3, and 5.

The final answer is $\boxed{\text{a) 2, 3, 5}}$

4. The H.C.F. of 13 and 39 is ......
List the factors of each number.
Factors of 13: 1, 13 (13 is a prime number)
Factors of 39: 1, 3, 13, 39
Identify the common factors.
The common factors are 1 and 13.
Determine the highest common factor (H.C.F.).
The highest common factor is 13.

The final answer is $\boxed{\text{d) 13}}$

5. Which of the following is the least common multiple of 6, 5 and 4?
Find the prime factorization of each number.
$6 = 2 \times 3$
$5 = 5$ (5 is a prime number)
$4 = 2 \times 2 = 2^2$
To find the L.C.M., take the highest power of each prime factor present in any of the numbers.
The prime factors are 2, 3, and 5.
Highest power of 2: $2^2$ (from 4)
Highest power of 3: $3^1$ (from 6)
Highest power of 5: $5^1$ (from 5)
Multiply these highest powers together.
$L.C.M. = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 12 \times 5 = 60$
The final answer is $\boxed{\text{c) 60}}$

6. What is $3.75 + 2.0001 + 6.3009$ ?
Align the decimal points and add the numbers.
3.7500 2.0001 + 6.3009 ---------
Perform the addition column by column, starting from the rightmost digit.
$0 + 1 + 9 = 10$ (write 0, carry 1)
$0 + 0 + 0 + 1 (\text{carry}) = 1$
$5 + 0 + 0 = 5$
$7 + 0 + 3 = 10$ (write 0, carry 1)
$3 + 2 + 6 + 1 (\text{carry}) = 12$
Combine the results.
$3.75 + 2.0001 + 6.3009 = 12.0510$
The final answer is $\boxed{\text{a) 12.510}}$

This is a set of math problems covering Roman numerals, subtraction, prime factorization, HCF, and LCM.

Step-by-step solution

Break down the Roman numeral into its components.

Add the values.

Set up the subtraction.

Perform the subtraction.

Find the prime factors of 30.

List the prime factors.

List the factors of each number.

Identify the common factors.

Determine the highest common factor (H.C.F.).

Find the prime factorization of each number.

To find the L.C.M., take the highest power of each prime factor present in any of the numbers.

Multiply these highest powers together.

Align the decimal points and add the numbers.

Perform the addition column by column, starting from the rightmost digit.

Combine the results.

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