What is the least number which when divided by 15, 18 and 21 leaves remainders 2, 5 and 8 respectively?

Answer

To find the least number which, when divided by 15, 18, and 21 leaves remainders of 2, 5, and 8 respectively, we can follow these steps:

### Step 1: Understand the Problem
We need to find a number \( x \) such that:
- \( x \mod 15 = 2 \)
- \( x \mod 18 = 5 \)
- \( x \mod 21 = 8 \)

### Step 2: Rewrite the Conditions
We can rewrite these conditions in terms of \( x \):
- \( x = 15k + 2 \) for some integer \( k \)
- \( x = 18m + 5 \) for some integer \( m \)
- \( x = 21n + 8 \) for some integer \( n \)

### Step 3: Adjust the Equations
To make the calculations easier, we can adjust each equation to find a common form:
- From \( x \mod 15 = 2 \), we can write \( x - 2 = 15k \)
- From \( x \mod 18 = 5 \), we can write \( x - 5 = 18m \)
- From \( x \mod 21 = 8 \), we can write \( x - 8 = 21n \)

### Step 4: Find the Differences
Now, we can find the differences between the numbers and their respective remainders:
- \( 15 - 2 = 13 \)
- \( 18 - 5 = 13 \)
- \( 21 - 8 = 13 \)

This shows that the differences are equal, which means we can find a common multiple.

### Step 5: Calculate the LCM
Next, we need to calculate the Least Common Multiple (LCM) of the divisors:
- The numbers we need to find the LCM for are 15, 18, and 21.

#### Finding the LCM:
1. **Prime Factorization**:
- \( 15 = 3 \times 5 \)
- \( 18 = 2 \times 3^2 \)
- \( 21 = 3 \times 7 \)

2. **Take the highest power of each prime**:
- From \( 2 \): \( 2^1 \)
- From \( 3 \): \( 3^2 \)
- From \( 5 \): \( 5^1 \)
- From \( 7 \): \( 7^1 \)

3. **Calculate the LCM**:
\[ \text{LCM} = 2^1 \times 3^2 \times 5^1 \times 7^1 = 2 \times 9 \times 5 \times 7 \]
\[ = 2 \times 9 = 18 \]
\[ = 18 \times 5 = 90 \]
\[ = 90 \times 7 = 630 \]

### Step 6: Add the Remainder
Now, since we are looking for the least number \( x \) that satisfies the original conditions, we add the common remainder (which is 13) to the LCM:
\[ x = 630 + 2 = 632 \]

### Final Answer
Thus, the least number which when divided by 15, 18, and 21 leaves remainders 2, 5, and 8 respectively is **632**.

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