What is the least number which when divided by 36 , 38 , 57 , 114 and 19 leaves the remainder 26 , 28 , 47 , 104 and 9 respectively ?

Answer

To solve the problem of finding the least number which, when divided by 36, 38, 57, 114, and 19, leaves the remainders 26, 28, 47, 104, and 9 respectively, we can follow these steps:

### Step 1: Set up the equations
We know that for each divisor \( d \) and remainder \( r \), the relationship can be expressed as:
\[ X = d \cdot q + r \]
where \( q \) is the quotient. We can rearrange this to:
\[ X - r = d \cdot q \]
This means \( X - r \) must be a multiple of \( d \).

### Step 2: Write down the equations
From the information given:
1. \( X - 26 = 36k \) (for some integer \( k \))
2. \( X - 28 = 38m \) (for some integer \( m \))
3. \( X - 47 = 57n \) (for some integer \( n \))
4. \( X - 104 = 114p \) (for some integer \( p \))
5. \( X - 9 = 19q \) (for some integer \( q \))

### Step 3: Find the differences
From the equations, we can express \( X \) in terms of the remainders:
- \( X = 36k + 26 \)
- \( X = 38m + 28 \)
- \( X = 57n + 47 \)
- \( X = 114p + 104 \)
- \( X = 19q + 9 \)

### Step 4: Calculate the common differences
We can observe that the differences between each divisor and its respective remainder are:
- \( 36 - 26 = 10 \)
- \( 38 - 28 = 10 \)
- \( 57 - 47 = 10 \)
- \( 114 - 104 = 10 \)
- \( 19 - 9 = 10 \)

This indicates that \( X - r \) is consistently 10 less than each divisor.

### Step 5: Find the LCM of the divisors
Now, we need to find the least common multiple (LCM) of the divisors: 36, 38, 57, 114, and 19.

Using the prime factorization method:
- \( 36 = 2^2 \times 3^2 \)
- \( 38 = 2 \times 19 \)
- \( 57 = 3 \times 19 \)
- \( 114 = 2 \times 3 \times 19 \)
- \( 19 = 19^1 \)

The LCM is found by taking the highest power of each prime:
- \( 2^2 \) from 36
- \( 3^2 \) from 36
- \( 19^1 \) from any of the numbers

Thus, the LCM is:
\[ \text{LCM} = 2^2 \times 3^2 \times 19 = 4 \times 9 \times 19 = 684 \]

### Step 6: Calculate the least number
Since we have established that \( X - r \) is equal to the LCM minus the common difference (10), we can calculate:
\[ X = \text{LCM} - 10 = 684 - 10 = 674 \]

### Final Answer
The least number \( X \) is **674**.

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