What is the least possible number which when divided by 21,25,27 and 35 it leaves the remainder 2 in each case ?

Answer

To find the least possible number which, when divided by 21, 25, 27, and 35, leaves a remainder of 2 in each case, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( N \) such that:
- \( N \mod 21 = 2 \)
- \( N \mod 25 = 2 \)
- \( N \mod 27 = 2 \)
- \( N \mod 35 = 2 \)

This means that \( N - 2 \) must be divisible by all these numbers.

### Step 2: Set Up the Equation
Let \( M = N - 2 \). Therefore, we need to find the least common multiple (LCM) of 21, 25, 27, and 35.

### Step 3: Factorize Each Number
- \( 21 = 3^1 \times 7^1 \)
- \( 25 = 5^2 \)
- \( 27 = 3^3 \)
- \( 35 = 5^1 \times 7^1 \)

### Step 4: Find the LCM
To find the LCM, take the highest power of each prime factor:
- For \( 3 \): the highest power is \( 3^3 \) (from 27)
- For \( 5 \): the highest power is \( 5^2 \) (from 25)
- For \( 7 \): the highest power is \( 7^1 \) (from both 21 and 35)

Thus, the LCM is:
\[ \text{LCM} = 3^3 \times 5^2 \times 7^1 \]

### Step 5: Calculate the LCM
Now, calculate the LCM:
\[ 3^3 = 27 \]
\[ 5^2 = 25 \]
\[ 7^1 = 7 \]
Now multiply these together:
\[ \text{LCM} = 27 \times 25 \times 7 \]

Calculating step-by-step:
1. \( 27 \times 25 = 675 \)
2. \( 675 \times 7 = 4725 \)

So, the LCM of 21, 25, 27, and 35 is \( 4725 \).

### Step 6: Find the Least Possible Number
Since \( M = N - 2 \), we have:
\[ M = 4725 \]
Thus,
\[ N = M + 2 = 4725 + 2 = 4727 \]

### Final Answer
The least possible number which when divided by 21, 25, 27, and 35 leaves a remainder of 2 is \( \boxed{4727} \).

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