Find the biggest number which divides 200 and 320 completely?

Options

A. 10
B. 20
C. 16
D. 40

Answer

To find the biggest number that divides both 200 and 320 completely, we need to calculate the Highest Common Factor (HCF) of the two numbers. We will use the factorization method to do this.

### Step-by-Step Solution:

1. **Factorization of 200**:
- Start by dividing 200 by the smallest prime number, which is 2.
- 200 ÷ 2 = 100
- 100 ÷ 2 = 50
- 50 ÷ 2 = 25
- 25 is not divisible by 2, so we move to the next prime number, which is 5.
- 25 ÷ 5 = 5
- 5 ÷ 5 = 1
- Therefore, the prime factorization of 200 is:
\[ 200 = 2^3 \times 5^2 \]

2. **Factorization of 320**:
- Start by dividing 320 by 2.
- 320 ÷ 2 = 160
- 160 ÷ 2 = 80
- 80 ÷ 2 = 40
- 40 ÷ 2 = 20
- 20 ÷ 2 = 10
- 10 ÷ 2 = 5
- 5 ÷ 5 = 1
- Therefore, the prime factorization of 320 is:
\[ 320 = 2^6 \times 5^1 \]

3. **Finding the HCF**:
- To find the HCF, we take the lowest power of all common prime factors from the factorizations of 200 and 320.
- The common prime factors are 2 and 5.
- For 2: The lowest power is \(2^3\) (from 200).
- For 5: The lowest power is \(5^1\) (from 320).
- Therefore, the HCF is:
\[ HCF = 2^3 \times 5^1 = 8 \times 5 = 40 \]

4. **Conclusion**:
- The biggest number which divides both 200 and 320 completely is **40**.