Find the smallest positive number divisible by all of the numbers from 1 to 12
2^3 * 3^2 * 5 * 7 * 11
To find the smallest positive number divisible by all of the numbers from 1 to 12, we can use the concept of the least common multiple (LCM). The LCM of a set of numbers is the smallest multiple that can be evenly divided by each number in that set.
To find the LCM of a set of numbers, we can begin by listing the prime factorizations of those numbers. The prime factorization of a number represents the product of its unique prime factors.
Let's find the prime factorizations of the numbers from 1 to 12:
1: 1 (no prime factors)
2: 2
3: 3
4: 2^2
5: 5
6: 2 * 3
7: 7
8: 2^3
9: 3^2
10: 2 * 5
11: 11
12: 2^2 * 3
Next, we identify the highest power of each prime factor that appears in any of the numbers' prime factorizations:
2^3, 3^2, 5, 7, 11
Finally, we multiply these highest powers together to find the LCM:
LCM = 2^3 * 3^2 * 5 * 7 * 11
= 8 * 9 * 5 * 7 * 11
= 27720
Therefore, the smallest positive number divisible by all of the numbers from 1 to 12 is 27720.