If `21168=2^axx3^bxx7^c`, where a, b and c are natural numbers, then what is the value of (4a - 5b + c)?

Options

A. 0
B. 1
C. 2
D. 3

Answer

To solve the equation \( 21168 = 2^a \times 3^b \times 7^c \), we need to find the prime factorization of 21168.

### Step 1: Prime Factorization of 21168
We start by dividing 21168 by the smallest prime number, which is 2.

- \( 21168 \div 2 = 10584 \)
- \( 10584 \div 2 = 5292 \)
- \( 5292 \div 2 = 2646 \)
- \( 2646 \div 2 = 1323 \)

Now, 1323 is not divisible by 2, so we move to the next prime number, which is 3.

- \( 1323 \div 3 = 441 \)
- \( 441 \div 3 = 147 \)
- \( 147 \div 3 = 49 \)

Now, 49 is not divisible by 3, so we move to the next prime number, which is 7.

- \( 49 \div 7 = 7 \)
- \( 7 \div 7 = 1 \)

Now we have completely factored 21168. The prime factorization is:

\[ 21168 = 2^4 \times 3^3 \times 7^2 \]

### Step 2: Identify the values of a, b, and c
From the prime factorization, we can identify:
- \( a = 4 \)
- \( b = 3 \)
- \( c = 2 \)

### Step 3: Calculate \( 4a - 5b + c \)
Now we can substitute the values of \( a \), \( b \), and \( c \) into the expression \( 4a - 5b + c \):

\[ 4a - 5b + c = 4(4) - 5(3) + 2 \]

Calculating this step-by-step:

1. Calculate \( 4a \):
\[ 4(4) = 16 \]

2. Calculate \( 5b \):
\[ 5(3) = 15 \]

3. Now substitute these values into the expression:
\[ 16 - 15 + 2 \]

4. Perform the subtraction and addition:
\[ 16 - 15 = 1 \]
\[ 1 + 2 = 3 \]

### Final Answer
Thus, the value of \( 4a - 5b + c \) is \( 3 \).