The Mayan Empire, which ruled Central America for centuries, had two calendars. One calendar was the same
as ours, 365 days long (without leap year). The other calendar was 270 days long, the length of a woman's
pregnancy. If both start on day one at the same time, then after how many of our years will the two calendars
begin at day one at the same time again?
This is like finding the LCM of the two numbers
365 = 5x73
270 = 5x2x27
LCM = 5x2x27x73 = 19710 days
or 54 of our year
To find out how many of our years it will take for the two calendars to begin at day one at the same time again, we need to determine the least common multiple (LCM) of 365 and 270.
The LCM is the smallest positive number that is divisible by both 365 and 270.
One way to find the LCM is by prime factorization.
Prime factorization of 365:
365 = 5 × 73
Prime factorization of 270:
270 = 2 × 3 × 3 × 3 × 5
To determine the LCM, we take the highest power of each prime factor that appears in either of the numbers:
2 (highest power is 1)
3 (highest power is 3)
5 (highest power is 1)
73 (highest power is 1)
Now we multiply these prime factors together:
2 × 3 × 3 × 3 × 5 × 73 = 39,690
Therefore, it will take 39,690 of our years for the two calendars to begin at day one at the same time again.