What is the largest number that divides each one of 1152 and 1664 exactly ?

Options

A. 32
B. 64
C. 128
D. 256

Explanation

To find the largest number that divides both 1152 and 1664 exactly, we need to calculate the Greatest Common Divisor (GCD) of the two numbers. We can do this using the prime factorization method.

### Step 1: Prime Factorization of 1152
1. Divide 1152 by 2 (the smallest prime number):
- 1152 ÷ 2 = 576
2. Divide 576 by 2:
- 576 ÷ 2 = 288
3. Divide 288 by 2:
- 288 ÷ 2 = 144
4. Divide 144 by 2:
- 144 ÷ 2 = 72
5. Divide 72 by 2:
- 72 ÷ 2 = 36
6. Divide 36 by 2:
- 36 ÷ 2 = 18
7. Divide 18 by 2:
- 18 ÷ 2 = 9
8. Now, divide 9 by 3 (the next prime number):
- 9 ÷ 3 = 3
9. Finally, divide 3 by 3:
- 3 ÷ 3 = 1

So, the prime factorization of 1152 is:
\[ 1152 = 2^7 \times 3^2 \]

### Step 2: Prime Factorization of 1664
1. Divide 1664 by 2:
- 1664 ÷ 2 = 832
2. Divide 832 by 2:
- 832 ÷ 2 = 416
3. Divide 416 by 2:
- 416 ÷ 2 = 208
4. Divide 208 by 2:
- 208 ÷ 2 = 104
5. Divide 104 by 2:
- 104 ÷ 2 = 52
6. Divide 52 by 2:
- 52 ÷ 2 = 26
7. Divide 26 by 2:
- 26 ÷ 2 = 13
8. Now, divide 13 by 13 (the next prime number):
- 13 ÷ 13 = 1

So, the prime factorization of 1664 is:
\[ 1664 = 2^7 \times 13^1 \]

### Step 3: Finding the GCD
Now that we have the prime factorizations:
- \( 1152 = 2^7 \times 3^2 \)
- \( 1664 = 2^7 \times 13^1 \)

To find the GCD, we take the lowest power of all prime factors present in both factorizations:
- For the prime number 2, the minimum power is \( 2^7 \).
- The primes 3 and 13 are not common in both factorizations.

Thus, the GCD is:
\[ GCD = 2^7 = 128 \]

### Final Answer
The largest number that divides both 1152 and 1664 exactly is **128**.

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