A number, when divided by 8, 12 and 15, gives always remainder 5. Then number is:

Question

Accepted Answer

To determine why Option 2 (9605) is the correct answer, we need to understand the conditions given in the problem. The number, when divided by 8, 12, and 15, should always leave a remainder of 5. This means that the number can be expressed in the form:

$$ \text{Number} = \text{LCM}(8, 12, 15) \times k + 5 $$

where $ k $ is an integer, and LCM stands for the least common multiple.

**Step 1: Calculate the LCM of 8, 12, and 15**

- Prime factorization of 8: $ 2^3 $
- Prime factorization of 12: $ 2^2 \times 3 $
- Prime factorization of 15: $ 3 \times 5 $

The LCM is obtained by taking the highest power of each prime number that appears in the factorizations:

- Highest power of 2: $ 2^3 = 8 $
- Highest power of 3: $ 3^1 = 3 $
- Highest power of 5: $ 5^1 = 5 $

Thus, the LCM is:

$$ \text{LCM} = 8 \times 3 \times 5 = 120 $$

**Step 2: Formulate the expression for the number**

The number can be expressed as:

$$ \text{Number} = 120k + 5 $$

**Step 3: Verify each option**

1. **Option 1: 9505**
   - $ 9505 - 5 = 9500 $
   - $ 9500 \div 120 = 79.1667 $ (not an integer)

2. **Option 2: 9605**
   - $ 9605 - 5 = 9600 $
   - $ 9600 \div 120 = 80 $ (an integer)

3. **Option 3: 9705**
   - $ 9705 - 5 = 9700 $
   - $ 9700 \div 120 = 80.8333 $ (not an integer)

4. **Option 4: 9805**
   - $ 9805 - 5 = 9800 $
   - $ 9800 \div 120 = 81.6667 $ (not an integer)

**Conclusion:**

Option 2 (9605) is the only number that, when reduced by 5, results in a multiple of 120. Therefore, it satisfies the condition of leaving a remainder of 5 when divided by 8, 12, and 15. This confirms that Option 2 is the correct answer.