To solve the problem step by step, we need to find a number that meets the following criteria:
1. When divided by 6, 7, 15, 20, and 21, it leaves a remainder of 1.
2. It is completely divisible by 29.
### Step 1: Find the LCM of the numbers 6, 7, 15, 20, and 21.
To find the least common multiple (LCM), we can factor each number into its prime factors:
- 6 = 2 × 3
- 7 = 7
- 15 = 3 × 5
- 20 = 2^2 × 5
- 21 = 3 × 7
Now, we take the highest power of each prime number that appears in these factorizations:
- The highest power of 2 is 2^2 (from 20).
- The highest power of 3 is 3^1 (from 6, 15, and 21).
- The highest power of 5 is 5^1 (from 15 and 20).
- The highest power of 7 is 7^1 (from 7 and 21).
Now, we can calculate the LCM:
\[
\text{LCM} = 2^2 \times 3^1 \times 5^1 \times 7^1 = 4 \times 3 \times 5 \times 7
\]
Calculating step by step:
- \(4 \times 3 = 12\)
- \(12 \times 5 = 60\)
- \(60 \times 7 = 420\)
So, the LCM of 6, 7, 15, 20, and 21 is **420**.
### Step 2: Formulate the required number.
Since the number leaves a remainder of 1 when divided by these numbers, we can express the required number as:
\[
\text{Required Number} = 420k + 1
\]
where \(k\) is a positive integer.
### Step 3: Ensure the number is divisible by 29.
Now we need to find \(k\) such that \(420k + 1\) is divisible by 29. This means:
\[
420k + 1 \equiv 0 \mod 29
\]
This can be rewritten as:
\[
420k \equiv -1 \mod 29
\]
First, we need to find \(420 \mod 29\):
Calculating \(420 \div 29\):
\[
420 \div 29 \approx 14.4827586 \quad \text{(take the integer part, which is 14)}
\]
\[
29 \times 14 = 406
\]
\[
420 - 406 = 14
\]
So, \(420 \equiv 14 \mod 29\).
Now we need to solve:
\[
14k \equiv -1 \mod 29
\]
This is equivalent to:
\[
14k \equiv 28 \mod 29
\]
To find \(k\), we can multiply both sides by the modular inverse of 14 modulo 29. The modular inverse can be found using the Extended Euclidean Algorithm, or we can test small values of \(k\):
Testing \(k = 2\):
\[
14 \times 2 = 28 \equiv 28 \mod 29 \quad \text{(this works)}
\]
### Step 4: Calculate the required number.
Now substitute \(k = 2\) back into the equation for the required number:
\[
\text{Required Number} = 420 \times 2 + 1 = 840 + 1 = 841
\]
Thus, the required number is **841**.
### Summary of the Solution:
The number that meets the criteria is **841**.
