All questions of Factors for Class 10 Exam

Prime Factors of 84:
To determine the prime factors of 84, we need to find the prime numbers that divide evenly into 84. A prime number is a number greater than 1 that is only divisible by 1 and itself.

Step 1: Prime factorization of 84:
To begin, we can start by dividing 84 by the smallest prime number, which is 2.

84 ÷ 2 = 42

Now, we have a new number, 42. We continue dividing by 2 until we can no longer divide evenly.

42 ÷ 2 = 21

Again, we have a new number, 21. We continue dividing by 2.

21 ÷ 2 = 10.5 (not divisible evenly)

2 is the only prime factor of 84 that we have found so far.

Step 2: Continue with the next prime number:
Since 2 is not a factor of 21, we move on to the next prime number, which is 3.

21 ÷ 3 = 7

Now, we have a new number, 7. We continue dividing by 3.

7 ÷ 3 = 2.33 (not divisible evenly)

3 is another prime factor of 84 that we have found.

Step 3: Final prime factor:
Since 3 is not a factor of 7, we move on to the next prime number, which is 5.

7 ÷ 5 = 1.4 (not divisible evenly)

5 is not a prime factor of 84.

Summary:
In conclusion, the prime factors of 84 are 2 and 3. Thus, the correct answer is option A, which is 2.

Correct answer is C

The least common multiple (LCM) is the smallest number that is divisible by all the given numbers. To find the LCM of 8, 12, and 16, we can use the prime factorization method.

1. Prime factorization of each number:
- 8 = 2^3
- 12 = 2^2 * 3
- 16 = 2^4

2. Identify the highest exponent for each prime factor:
- 2^4 (since 16 has the highest exponent of 4)
- 3 (since 12 has an exponent of 1)
- No other prime factors present

3. Multiply the prime factors with their highest exponents:
2^4 * 3 = 16 * 3 = 48

Therefore, the LCM of 8, 12, and 16 is 48, which corresponds to option B.

The greatest common factor (GCF) of three numbers is the largest number that divides evenly into all three numbers. To find the GCF of 42, 56, and 70, we can use several methods, including prime factorization and listing factors.

Prime Factorization Method:
1. Start by finding the prime factorization of each number:
- 42 = 2 × 3 × 7
- 56 = 2 × 2 × 2 × 7
- 70 = 2 × 5 × 7

2. Identify the common prime factors among the three numbers. In this case, the only common prime factor is 7.

3. Multiply the common prime factors to find the GCF: 7.

Listing Factors Method:
1. List the factors of each number:
- Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
- Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56
- Factors of 70: 1, 2, 5, 7, 10, 14, 35, 70

2. Identify the common factors among the three numbers. In this case, the only common factor is 7.

3. The largest common factor is 7.

Therefore, the greatest common factor of 42, 56, and 70 is 7, which is option B.

The smallest multiple of both 6 and 10 can be found by finding the least common multiple (LCM) of the two numbers. The LCM is the smallest positive integer that is divisible by both numbers.

To find the LCM of 6 and 10, we can list their multiples and look for the first common multiple:

Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ...

From the lists above, we can see that the first common multiple is 30. Therefore, the smallest multiple of both 6 and 10 is 30.

Explanation:
- The question asks for the smallest multiple of both 6 and 10, so we need to find their least common multiple (LCM).
- The LCM is the smallest positive integer that is divisible by both numbers.
- To find the LCM, we can list the multiples of each number and look for the first common multiple.
- The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, ...
- The multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, ...
- From the lists above, we can see that the first common multiple is 30.
- Therefore, the smallest multiple of both 6 and 10 is 30.
- Option C, which is 30, is the correct answer.
- It is important to note that we can also find the LCM using prime factorization, but listing the multiples is a simpler method for small numbers like 6 and 10.

C

Factors of 36:
To find the factors of a number, we need to determine all the numbers that can divide the given number without leaving a remainder. Let's find the factors of 36.

1. Prime factorization of 36:
To find the prime factorization of 36, we need to express it as a product of prime numbers. The prime factors of 36 are 2 and 3.

36 = 2^2 * 3^2

2. Finding the factors:
To find the factors of 36, we can list all the possible combinations of the prime factors.

Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36

Explanation:
- 1 is always a factor of any number.
- 2 is a factor because 2^2 is a factor of 36.
- 3 is a factor because 3^2 is a factor of 36.
- 4 is a factor because it is a combination of 2^2 and 3^0 (any number raised to the power of 0 is 1).
- Similarly, 6 is a factor because it is a combination of 2^1 and 3^1.
- 9 is a factor because it is a combination of 2^0 and 3^2.
- 12 is a factor because it is a combination of 2^2 and 3^1.
- 18 is a factor because it is a combination of 2^1 and 3^2.
- Finally, 36 is a factor of itself.

Therefore, the number 36 has a total of 9 factors.