For every question, four probable answers bear in letters (A), (B), (C) and (D) are given. Only one out of these is correct. You have to choose the correct answer.
The greatest number that will divide 480 and 840 exactly is

Options

A. 110
B. 115
C. 120
D. 125

Answer

To find the greatest number that will divide both 480 and 840 exactly, we need to determine the greatest common divisor (GCD) of these two numbers. Here’s how we can solve the problem step by step:

### Step 1: Find the prime factorization of 480.
- We can divide 480 by 2 (the smallest prime number):
- 480 ÷ 2 = 240
- 240 ÷ 2 = 120
- 120 ÷ 2 = 60
- 60 ÷ 2 = 30
- 30 ÷ 2 = 15
- 15 ÷ 3 = 5
- 5 ÷ 5 = 1
- Therefore, the prime factorization of 480 is:
\[ 480 = 2^5 \times 3^1 \times 5^1 \]

### Step 2: Find the prime factorization of 840.
- We can divide 840 by 2:
- 840 ÷ 2 = 420
- 420 ÷ 2 = 210
- 210 ÷ 2 = 105
- 105 ÷ 3 = 35
- 35 ÷ 5 = 7
- 7 ÷ 7 = 1
- Therefore, the prime factorization of 840 is:
\[ 840 = 2^3 \times 3^1 \times 5^1 \times 7^1 \]

### Step 3: Identify the common prime factors.
- From the prime factorizations:
- 480: \(2^5 \times 3^1 \times 5^1\)
- 840: \(2^3 \times 3^1 \times 5^1 \times 7^1\)
- The common prime factors are:
- For 2: the minimum power is \(2^3\)
- For 3: the minimum power is \(3^1\)
- For 5: the minimum power is \(5^1\)

### Step 4: Calculate the GCD using the common prime factors.
- The GCD is calculated as:
\[ GCD = 2^3 \times 3^1 \times 5^1 \]
- Now, calculate the value:
\[ GCD = 8 \times 3 \times 5 = 120 \]

### Conclusion:
- The greatest number that will divide both 480 and 840 exactly is **120**.