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?
To find out after how many years the two calendars will begin at day one at the same time again, we need to determine the least common multiple (LCM) of the two calendar lengths.
The first calendar has a length of 365 days, while the second calendar has a length of 270 days. To find the LCM, we need to determine the smallest number that is divisible by both 365 and 270.
To find the LCM, we can use the prime factorization method. We will break down 365 and 270 into their prime factors, and then take the highest power of each prime factor.
Prime factorization of 365:
365 = 5 x 73
Prime factorization of 270:
270 = 2 x 3 x 3 x 3 x 5
Now, let's take the highest power of each prime factor:
Highest power of 2: 2^1
Highest power of 3: 3^3
Highest power of 5: 5^1
Highest power of 73: 73^1
Now, we can multiply these highest powers together to get the LCM:
LCM = 2^1 x 3^3 x 5^1 x 73^1 = 2 x 27 x 5 x 73 = 19,710
Therefore, the two calendars will begin at day one at the same time again after 19,710 of our years.