To find the least number which, when divided by 12, 18, 36, and 45, leaves the remainders 8, 14, 32, and 41 respectively, we can follow these steps:
### Step 1: Set up the equations based on the remainders
We know that if a number \( x \) leaves a remainder when divided by another number, we can express this as:
- \( x \equiv 8 \mod 12 \)
- \( x \equiv 14 \mod 18 \)
- \( x \equiv 32 \mod 36 \)
- \( x \equiv 41 \mod 45 \)
### Step 2: Rewrite the equations
We can rewrite these equations in terms of \( x \):
- \( x = 12k + 8 \) for some integer \( k \)
- \( x = 18m + 14 \) for some integer \( m \)
- \( x = 36n + 32 \) for some integer \( n \)
- \( x = 45p + 41 \) for some integer \( p \)
### Step 3: Find the differences between the divisors and the remainders
Now, we calculate the differences between each divisor and its corresponding remainder:
- \( 12 - 8 = 4 \)
- \( 18 - 14 = 4 \)
- \( 36 - 32 = 4 \)
- \( 45 - 41 = 4 \)
This shows that all the differences are equal to 4. This means that \( x - 4 \) is divisible by all the divisors (12, 18, 36, and 45).
### Step 4: Find the LCM of the divisors
Next, we need to find the least common multiple (LCM) of the numbers 12, 18, 36, and 45.
**Finding LCM:**
1. Prime factorization:
- \( 12 = 2^2 \times 3^1 \)
- \( 18 = 2^1 \times 3^2 \)
- \( 36 = 2^2 \times 3^2 \)
- \( 45 = 3^2 \times 5^1 \)
2. Take the highest power of each prime:
- For \( 2 \): \( 2^2 \)
- For \( 3 \): \( 3^2 \)
- For \( 5 \): \( 5^1 \)
3. Calculate the LCM:
\[
\text{LCM} = 2^2 \times 3^2 \times 5^1 = 4 \times 9 \times 5 = 180
\]
### Step 5: Calculate the least number
Since \( x - 4 \) is divisible by 180, we can express this as:
\[
x - 4 = 180k \quad \text{for some integer } k
\]
Thus,
\[
x = 180k + 4
\]
### Step 6: Find the least value of \( x \)
To find the least number, we can set \( k = 1 \):
\[
x = 180 \times 1 + 4 = 184
\]
### Step 7: Verify the solution
Now we check if 184 gives the correct remainders:
- \( 184 \div 12 = 15 \) remainder \( 8 \)
- \( 184 \div 18 = 10 \) remainder \( 14 \)
- \( 184 \div 36 = 5 \) remainder \( 32 \)
- \( 184 \div 45 = 4 \) remainder \( 41 \)
All remainders match the conditions given in the problem.
### Final Answer
The least number which when divided by 12, 18, 36, and 45 leaves the remainders 8, 14, 32, and 41 respectively is **184**.
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