A number is divisible by 7 and 12. Find all the other numbers which divide the given number

Answer

To solve the problem step by step, we need to find a number that is divisible by both 7 and 12, and then identify all the other numbers that divide this number.

### Step 1: Identify the numbers
The numbers we are working with are 7 and 12.

### Step 2: Find the factors of each number
- The factors of 7 are: 1, 7
- The factors of 12 are: 1, 2, 3, 4, 6, 12

### Step 3: Find the Least Common Multiple (LCM)
To find the LCM of 7 and 12:
- The prime factorization of 7 is simply 7 (since it is a prime number).
- The prime factorization of 12 is: \( 2^2 \times 3^1 \).

Now, we take the highest power of each prime factor:
- From 7: \( 7^1 \)
- From 12: \( 2^2 \) and \( 3^1 \)

Thus, the LCM is:
\[ LCM = 7^1 \times 2^2 \times 3^1 = 7 \times 4 \times 3 \]
Calculating this gives:
\[ 7 \times 4 = 28 \]
\[ 28 \times 3 = 84 \]

### Step 4: Identify all divisors of the LCM (84)
To find all the numbers that divide 84, we can list the factors of 84. The factors of 84 can be found by dividing 84 by integers starting from 1 up to 84.

The factors of 84 are:
1. \( 84 \div 1 = 84 \)
2. \( 84 \div 2 = 42 \)
3. \( 84 \div 3 = 28 \)
4. \( 84 \div 4 = 21 \)
5. \( 84 \div 6 = 14 \)
6. \( 84 \div 7 = 12 \)
7. \( 84 \div 12 = 7 \)
8. \( 84 \div 14 = 6 \)
9. \( 84 \div 21 = 4 \)
10. \( 84 \div 28 = 3 \)
11. \( 84 \div 42 = 2 \)
12. \( 84 \div 84 = 1 \)

Thus, the complete list of divisors of 84 is:
1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, 84.

### Final Answer
The numbers that divide the given number (which is 84) are:
1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42, and 84.

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