To solve the problem, we need to find the least number \( x \) that satisfies the following conditions:
1. \( x \) is divisible by 13.
2. When \( x \) is divided by 4, 5, 6, 7, 8, and 12, the remainder is 2.
### Step 1: Find the Least Common Multiple (LCM)
First, we need to find the least common multiple (LCM) of the numbers 4, 5, 6, 7, 8, and 12.
- The prime factorization of each number is:
- \( 4 = 2^2 \)
- \( 5 = 5^1 \)
- \( 6 = 2^1 \times 3^1 \)
- \( 7 = 7^1 \)
- \( 8 = 2^3 \)
- \( 12 = 2^2 \times 3^1 \)
To find the LCM, we take the highest power of each prime:
- For \( 2 \): \( 2^3 \) (from 8)
- For \( 3 \): \( 3^1 \) (from 6 or 12)
- For \( 5 \): \( 5^1 \) (from 5)
- For \( 7 \): \( 7^1 \) (from 7)
Thus, the LCM is:
\[
\text{LCM} = 2^3 \times 3^1 \times 5^1 \times 7^1 = 8 \times 3 \times 5 \times 7
\]
Calculating this step-by-step:
\[
8 \times 3 = 24
\]
\[
24 \times 5 = 120
\]
\[
120 \times 7 = 840
\]
So, the LCM of 4, 5, 6, 7, 8, and 12 is \( 840 \).
### Step 2: Set Up the Equation
Since \( x \) leaves a remainder of 2 when divided by these numbers, we can express \( x \) as:
\[
x = 840k + 2
\]
for some integer \( k \).
### Step 3: Find the Condition for Divisibility by 13
Now, we also need \( x \) to be divisible by 13:
\[
840k + 2 \equiv 0 \mod{13}
\]
This simplifies to:
\[
840k \equiv -2 \mod{13}
\]
Calculating \( 840 \mod{13} \):
\[
840 \div 13 = 64 \quad \text{(the quotient)}
\]
\[
840 - (64 \times 13) = 840 - 832 = 8
\]
Thus,
\[
840 \equiv 8 \mod{13}
\]
So we have:
\[
8k \equiv -2 \mod{13}
\]
This can be rewritten as:
\[
8k \equiv 11 \mod{13}
\]
(Here, \(-2\) is equivalent to \(11\) in modulo \(13\)).
### Step 4: Solve for \( k \)
Now we need to find the multiplicative inverse of \( 8 \mod{13} \). We can check values:
- \( 8 \times 5 = 40 \equiv 1 \mod{13} \)
Thus, the inverse is \( 5 \). Now we can multiply both sides of the equation \( 8k \equiv 11 \) by \( 5 \):
\[
k \equiv 55 \mod{13}
\]
Calculating \( 55 \mod{13} \):
\[
55 \div 13 = 4 \quad \text{(the quotient)}
\]
\[
55 - (4 \times 13) = 55 - 52 = 3
\]
Thus,
\[
k \equiv 3 \mod{13}
\]
So, the smallest positive value for \( k \) is \( k = 3 \).
### Step 5: Calculate \( x \)
Now substituting \( k = 3 \) back into the equation for \( x \):
\[
x = 840 \times 3 + 2 = 2520 + 2 = 2522
\]
### Step 6: Find the Sum of the Digits of \( x \)
Now we need to find the sum of the digits of \( 2522 \):
\[
2 + 5 + 2 + 2 = 11
\]
### Final Answer
The sum of the digits of \( x \) is \( 11 \).
