To solve the problem, we need to find the least multiple of 29, denoted as \( x \), that satisfies the following conditions when divided by 20, 21, 22, 24, and 28:
- Remainder when divided by 20 is 13
- Remainder when divided by 21 is 14
- Remainder when divided by 22 is 15
- Remainder when divided by 24 is 17
- Remainder when divided by 28 is 21
### Step-by-Step Solution
1. **Set Up the Congruences**:
We can express the conditions in terms of congruences:
\[
\begin{align*}
x & \equiv 13 \pmod{20} \\
x & \equiv 14 \pmod{21} \\
x & \equiv 15 \pmod{22} \\
x & \equiv 17 \pmod{24} \\
x & \equiv 21 \pmod{28}
\end{align*}
\]
2. **Adjust the Congruences**:
To simplify, we can rewrite each congruence:
\[
\begin{align*}
x & \equiv -7 \pmod{20} \\
x & \equiv -7 \pmod{21} \\
x & \equiv -7 \pmod{22} \\
x & \equiv -7 \pmod{24} \\
x & \equiv -7 \pmod{28}
\end{align*}
\]
This shows that \( x + 7 \) is a common multiple of 20, 21, 22, 24, and 28.
3. **Find the LCM**:
Calculate the least common multiple (LCM) of the divisors:
\[
\text{LCM}(20, 21, 22, 24, 28) = 9240
\]
4. **Express \( x \)**:
Therefore, we can express \( x \) as:
\[
x = 9240k - 7
\]
for some integer \( k \).
5. **Find the Least Multiple of 29**:
We need \( x \) to be a multiple of 29:
\[
9240k - 7 \equiv 0 \pmod{29}
\]
This can be rearranged to:
\[
9240k \equiv 7 \pmod{29}
\]
6. **Calculate \( 9240 \mod 29 \)**:
First, we find \( 9240 \mod 29 \):
\[
9240 \div 29 \approx 318 \quad \text{(integer part)}
\]
\[
29 \times 318 = 9222 \quad \Rightarrow \quad 9240 - 9222 = 18
\]
Thus, \( 9240 \equiv 18 \pmod{29} \).
7. **Set Up the Equation**:
Now we have:
\[
18k \equiv 7 \pmod{29}
\]
8. **Find the Inverse of 18 Modulo 29**:
We need to find the multiplicative inverse of 18 modulo 29. Using the Extended Euclidean Algorithm, we find that the inverse is 25 (since \( 18 \times 25 \equiv 1 \pmod{29} \)).
9. **Solve for \( k \)**:
Multiply both sides of the congruence by 25:
\[
k \equiv 25 \times 7 \pmod{29} \equiv 175 \pmod{29} \equiv 1 \pmod{29}
\]
Thus, \( k = 1 \) is the smallest positive integer.
10. **Calculate \( x \)**:
Substitute \( k \) back into the equation for \( x \):
\[
x = 9240 \times 1 - 7 = 9233
\]
11. **Check if \( x \) is a Multiple of 29**:
Verify that \( 9233 \) is a multiple of 29:
\[
9233 \div 29 = 318 \quad \text{(exact division)}
\]
12. **Sum of Digits**:
Finally, we calculate the sum of the digits of \( 9233 \):
\[
9 + 2 + 3 + 3 = 17
\]
### Final Answer
The sum of the digits of \( x \) is **17**.
