The factors of a number include 9, 12, and 20. What is the smallest number this could be? List all the factors of this number
The smallest number this could be is 180.
Factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180
To find the smallest number that has factors of 9, 12, and 20, we need to find the least common multiple (LCM) of these numbers.
First, let's find the prime factorization of each number:
9 = 3 * 3
12 = 2 * 2 * 3
20 = 2 * 2 * 5
Now, we take the highest power of each prime factor that appears in any of the numbers:
2^2 * 3^2 * 5 = 4 * 9 * 5 = 180
Therefore, the smallest number that has factors of 9, 12, and 20 is 180.
Now, let's list all the factors of 180:
Factors of 180: 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.
To find the smallest number that has 9, 12, and 20 as factors, we need to find the least common multiple (LCM) of these three numbers. The LCM is the smallest multiple that is divisible by all of the given numbers.
To find the LCM of 9, 12, and 20, we can use the prime factorization method:
1. Start by writing the prime factorization of each number:
- 9 = 3 * 3
- 12 = 2 * 2 * 3
- 20 = 2 * 2 * 5
2. Identify the highest power of each prime factor used in any of the numbers. In this case, the factors are 2, 3, and 5. The highest power of 2 is 2^2 (from 12 and 20), the highest power of 3 is 3^2 (from 9), and the highest power of 5 is 5^1 (from 20).
3. Multiply the highest powers of the prime factors: 2^2 * 3^2 * 5^1 = 4 * 9 * 5 = 180.
Therefore, the smallest number that has 9, 12, and 20 as factors is 180.
Now let's list all the factors of 180 by finding all the numbers that divide it without leaving a remainder:
- 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180.
These are all the factors of the number 180.