Find the greatest number which will exactly divide 200 and 320.

Options

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

Answer

To find the greatest number that will exactly divide both 200 and 320, we need to determine the Highest Common Factor (HCF) or Greatest Common Divisor (GCD) of the two numbers. We can do this using prime factorization.

### Step 1: Prime Factorization of 200
1. Divide 200 by 2 (the smallest prime number):
- \( 200 \div 2 = 100 \)
2. Divide 100 by 2:
- \( 100 \div 2 = 50 \)
3. Divide 50 by 2:
- \( 50 \div 2 = 25 \)
4. Now, 25 is not divisible by 2, so we move to the next prime number, which is 5:
- \( 25 \div 5 = 5 \)
5. Finally, divide 5 by 5:
- \( 5 \div 5 = 1 \)

So, the prime factorization of 200 is:
\[ 200 = 2^3 \times 5^2 \]

### Step 2: Prime Factorization of 320
1. Divide 320 by 2:
- \( 320 \div 2 = 160 \)
2. Divide 160 by 2:
- \( 160 \div 2 = 80 \)
3. Divide 80 by 2:
- \( 80 \div 2 = 40 \)
4. Divide 40 by 2:
- \( 40 \div 2 = 20 \)
5. Divide 20 by 2:
- \( 20 \div 2 = 10 \)
6. Divide 10 by 2:
- \( 10 \div 2 = 5 \)
7. Finally, divide 5 by 5:
- \( 5 \div 5 = 1 \)

So, the prime factorization of 320 is:
\[ 320 = 2^6 \times 5^1 \]

### Step 3: Finding the HCF
Now, we will find the HCF by taking the lowest power of all prime factors present in both factorizations.

- For the prime number 2:
- In 200, the power is 3 (from \( 2^3 \)).
- In 320, the power is 6 (from \( 2^6 \)).
- The minimum power is \( 2^3 \).

- For the prime number 5:
- In 200, the power is 2 (from \( 5^2 \)).
- In 320, the power is 1 (from \( 5^1 \)).
- The minimum power is \( 5^1 \).

Now, we multiply these together to find the HCF:
\[ HCF = 2^3 \times 5^1 = 8 \times 5 = 40 \]

### Conclusion
The greatest number which will exactly divide both 200 and 320 is **40**.

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