The least number, which when divided by 18, 27 and 36 separately leaves remainders 5, 14, 23 respectively, is:

Options

A. 95
B. 113
C. 149
D. 77

Answer

To find the least number which, when divided by 18, 27, and 36, leaves remainders 5, 14, and 23 respectively, we can follow these steps:

### Step 1: Determine the negative remainders
For each divisor, we will calculate the negative remainder:
- For 18:
\[ \text{Negative Remainder} = 18 - 5 = 13 \]
- For 27:
\[ \text{Negative Remainder} = 27 - 14 = 13 \]
- For 36:
\[ \text{Negative Remainder} = 36 - 23 = 13 \]

### Step 2: Confirm the negative remainders
We can see that the negative remainder is the same for all three cases, which is 13.

### Step 3: Find the LCM of the divisors
Next, we will find the Least Common Multiple (LCM) of 18, 27, and 36.
- The prime factorization of the numbers is:
- \( 18 = 2 \times 3^2 \)
- \( 27 = 3^3 \)
- \( 36 = 2^2 \times 3^2 \)

To find the LCM, we take the highest power of each prime factor:
- For \( 2 \): highest power is \( 2^2 \) (from 36)
- For \( 3 \): highest power is \( 3^3 \) (from 27)

Thus, the LCM is:
\[ \text{LCM} = 2^2 \times 3^3 = 4 \times 27 = 108 \]

### Step 4: Subtract the negative remainder from the LCM
Now we subtract the negative remainder (13) from the LCM (108):
\[ \text{Required Least Number} = 108 - 13 = 95 \]

### Final Answer
The least number which, when divided by 18, 27, and 36, leaves remainders 5, 14, and 23 respectively is:
\[ \boxed{95} \]

---