Let x be the least number which when divided by 8, 12, 20, 28, 35 leaves a remainder 5 in each case. What is the sum of digits of x ?

Q: Let x be the least number which when divided by 8, 12, 20, 28, 35 leaves a remainder 5 in each case. What is the sum of digits of x ?

To solve the problem, we need to find the least number $ x $ that leaves a remainder of 5 when divided by 8, 12, 20, 28, and 35. This means that $ x - 5 $ must be divisible by each of these numbers. Therefore, $ x - 5 $ is a common multiple of these numbers. 1. **Finding the Least Common Multiple (LCM):** To find the LCM of 8, 12, 20, 28, and 35, we first determine the prime factorization of each number: - $ 8 = 2^3 $ - $ 12 = 2^2 \times 3 $ - $ 20 = 2^2 \times 5 $ - $ 28 = 2^2 \times 7 $ - $ 35 = 5 \times 7 $ The LCM is found by taking the highest power of each prime number that appears in these factorizations: - Highest power of 2: $ 2^3 $ - Highest power of 3: $ 3^1 $ - Highest power of 5: $ 5^1 $ - Highest power of 7: $ 7^1 $ Therefore, the LCM is: \[ 2^3 \times 3^1 \times 5^1 \times 7^1 = 8 \times 3 \times 5 \times 7 = 840 \] 2. **Finding $ x $:** Since $ x - 5 $ must be a multiple of 840, we have: \[ x - 5 = 840k \quad \text{for some integer } k \] \[ x = 840k + 5 \] The smallest positive $ x $ occurs when $ k = 1 $: \[ x = 840 \times 1 + 5 = 845 \] 3. **Sum of the Digits of $ x $:** Now, we calculate the sum of the digits of 845: \[ 8 + 4 + 5 = 17 \] Thus, the sum of the digits of $ x $ is 17, which corresponds to Option 4. **Explanation of Incorrect Options:** - **Option 1 (11):** The sum of the digits of 845 is not 11. - **Option 2 (14):** The sum of the digits of 845 is not 14. - **Option 3 (15):** The sum of the digits of 845 is not 15. Therefore, the correct answer is Option 4, as the sum of the digits of 845 is indeed 17.

Question

Accepted Answer

To solve the problem, we need to find the least number $ x $ that leaves a remainder of 5 when divided by 8, 12, 20, 28, and 35. This means that $ x - 5 $ must be divisible by each of these numbers. Therefore, $ x - 5 $ is a common multiple of these numbers.

1. **Finding the Least Common Multiple (LCM):**

To find the LCM of 8, 12, 20, 28, and 35, we first determine the prime factorization of each number:
   - $ 8 = 2^3 $
   - $ 12 = 2^2 \times 3 $
   - $ 20 = 2^2 \times 5 $
   - $ 28 = 2^2 \times 7 $
   - $ 35 = 5 \times 7 $

The LCM is found by taking the highest power of each prime number that appears in these factorizations:
   - Highest power of 2: $ 2^3 $
   - Highest power of 3: $ 3^1 $
   - Highest power of 5: $ 5^1 $
   - Highest power of 7: $ 7^1 $

Therefore, the LCM is:
   $$
   2^3 \times 3^1 \times 5^1 \times 7^1 = 8 \times 3 \times 5 \times 7 = 840
   $$

2. **Finding $ x $:**

Since $ x - 5 $ must be a multiple of 840, we have:
   $$
   x - 5 = 840k \quad \text{for some integer } k
   $$
   $$
   x = 840k + 5
   $$

The smallest positive $ x $ occurs when $ k = 1 $:
   $$
   x = 840 \times 1 + 5 = 845
   $$

3. **Sum of the Digits of $ x $:**

Now, we calculate the sum of the digits of 845:
   $$
   8 + 4 + 5 = 17
   $$

Thus, the sum of the digits of $ x $ is 17, which corresponds to Option 4.

**Explanation of Incorrect Options:**

- **Option 1 (11):** The sum of the digits of 845 is not 11.
- **Option 2 (14):** The sum of the digits of 845 is not 14.
- **Option 3 (15):** The sum of the digits of 845 is not 15.

Therefore, the correct answer is Option 4, as the sum of the digits of 845 is indeed 17.