What is the least possible number which when divided by 18, 35 or 42 it leaves the 2, 19, 26 as the remainders respectively?

Answer

To find the least possible number which, when divided by 18, 35, or 42, leaves the remainders 2, 19, and 26 respectively, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( N \) such that:
- \( N \mod 18 = 2 \)
- \( N \mod 35 = 19 \)
- \( N \mod 42 = 26 \)

### Step 2: Rewrite the Equations
From the modular equations, we can express \( N \) in terms of the divisors and remainders:
- \( N = 18k + 2 \) for some integer \( k \)
- \( N = 35m + 19 \) for some integer \( m \)
- \( N = 42n + 26 \) for some integer \( n \)

### Step 3: Find the LCM of the Divisors
Next, we need to find the least common multiple (LCM) of the divisors 18, 35, and 42.

1. **Prime Factorization**:
- \( 18 = 2^1 \times 3^2 \)
- \( 35 = 5^1 \times 7^1 \)
- \( 42 = 2^1 \times 3^1 \times 7^1 \)

2. **Finding LCM**:
- Take the highest power of each prime:
- \( 2^1 \) from 18 and 42
- \( 3^2 \) from 18
- \( 5^1 \) from 35
- \( 7^1 \) from 35 and 42

Therefore, the LCM is:
\[ LCM = 2^1 \times 3^2 \times 5^1 \times 7^1 = 2 \times 9 \times 5 \times 7 \]
\[ = 2 \times 9 = 18 \\ 18 \times 5 = 90 \\ 90 \times 7 = 630 \]

### Step 4: Adjust for Remainders
Now, since we need \( N \) to leave specific remainders, we can adjust our LCM:
- The differences between the divisors and their respective remainders are all equal to 16:
- \( 18 - 2 = 16 \)
- \( 35 - 19 = 16 \)
- \( 42 - 26 = 16 \)

Thus, we can express the least possible number \( N \) as:
\[ N = LCM - 16 = 630 - 16 = 614 \]

### Step 5: Conclusion
The least possible number which when divided by 18, 35, or 42 leaves the remainders 2, 19, and 26 respectively is:
\[ \boxed{614} \]