If 21168 = 2$^a$ × 3$^b$ × 7$^c$, where a, b and care natural numbers, then what is the value of (4a – 5b + c) ?

Q: If 21168 = 2$^a$ × 3$^b$ × 7$^c$, where a, b and care natural numbers, then what is the value of (4a – 5b + c) ?

To determine the value of $4a - 5b + c$ given that $21168 = 2^a \times 3^b \times 7^c$, we need to find the prime factorization of 21168. 1. **Prime Factorization of 21168:** - Divide 21168 by 2 (the smallest prime number): \[ 21168 \div 2 = 10584 \] - Divide 10584 by 2: \[ 10584 \div 2 = 5292 \] - Divide 5292 by 2: \[ 5292 \div 2 = 2646 \] - Divide 2646 by 2: \[ 2646 \div 2 = 1323 \] - 1323 is not divisible by 2. Check divisibility by 3: \[ 1323 \div 3 = 441 \] - Divide 441 by 3: \[ 441 \div 3 = 147 \] - Divide 147 by 3: \[ 147 \div 3 = 49 \] - 49 is not divisible by 3. Check divisibility by 7: \[ 49 \div 7 = 7 \] - Divide 7 by 7: \[ 7 \div 7 = 1 \] The prime factorization of 21168 is $2^4 \times 3^3 \times 7^2$. 2. **Values of $a$, $b$, and $c$:** - $a = 4$ (since the power of 2 is 4) - $b = 3$ (since the power of 3 is 3) - $c = 2$ (since the power of 7 is 2) 3. **Calculate $4a - 5b + c$:** \[ 4a - 5b + c = 4(4) - 5(3) + 2 = 16 - 15 + 2 = 3 \] Therefore, the value of $4a - 5b + c$ is 3, which corresponds to Option 4. **Explanation of Incorrect Options:** - **Option 1 (0):** This would imply that the expression $4a - 5b + c$ equals 0, which is incorrect based on our calculation. - **Option 2 (1):** This would imply that the expression $4a - 5b + c$ equals 1, which is incorrect based on our calculation. - **Option 3 (2):** This would imply that the expression $4a - 5b + c$ equals 2, which is incorrect based on our calculation. Thus, Option 4 is the correct answer, as it accurately reflects the calculated value of $4a - 5b + c$.

Question

Accepted Answer

To determine the value of $4a - 5b + c$ given that $21168 = 2^a \times 3^b \times 7^c$, we need to find the prime factorization of 21168.

1. **Prime Factorization of 21168:**
   - Divide 21168 by 2 (the smallest prime number):
     $$
     21168 \div 2 = 10584
     $$
   - Divide 10584 by 2:
     $$
     10584 \div 2 = 5292
     $$
   - Divide 5292 by 2:
     $$
     5292 \div 2 = 2646
     $$
   - Divide 2646 by 2:
     $$
     2646 \div 2 = 1323
     $$
   - 1323 is not divisible by 2. Check divisibility by 3:
     $$
     1323 \div 3 = 441
     $$
   - Divide 441 by 3:
     $$
     441 \div 3 = 147
     $$
   - Divide 147 by 3:
     $$
     147 \div 3 = 49
     $$
   - 49 is not divisible by 3. Check divisibility by 7:
     $$
     49 \div 7 = 7
     $$
   - Divide 7 by 7:
     $$
     7 \div 7 = 1
     $$

The prime factorization of 21168 is $2^4 \times 3^3 \times 7^2$.

2. **Values of $a$, $b$, and $c$:**
   - $a = 4$ (since the power of 2 is 4)
   - $b = 3$ (since the power of 3 is 3)
   - $c = 2$ (since the power of 7 is 2)

3. **Calculate $4a - 5b + c$:**
   $$
   4a - 5b + c = 4(4) - 5(3) + 2 = 16 - 15 + 2 = 3
   $$

Therefore, the value of $4a - 5b + c$ is 3, which corresponds to Option 4.

**Explanation of Incorrect Options:**
- **Option 1 (0):** This would imply that the expression $4a - 5b + c$ equals 0, which is incorrect based on our calculation.
- **Option 2 (1):** This would imply that the expression $4a - 5b + c$ equals 1, which is incorrect based on our calculation.
- **Option 3 (2):** This would imply that the expression $4a - 5b + c$ equals 2, which is incorrect based on our calculation.

Thus, Option 4 is the correct answer, as it accurately reflects the calculated value of $4a - 5b + c$.