How many Positive integers are between 200-500 that are divisible by the integers 4,6,10,12?
To find the number of positive integers between 200 and 500 that are divisible by 4, 6, 10, and 12, we can use the concept of the least common multiple (LCM) of these numbers.
Step 1: Find the LCM of 4, 6, 10, and 12.
The LCM of a set of numbers is the smallest number that all the given numbers can divide evenly into. In this case, the LCM of 4, 6, 10, and 12 is 60.
Step 2: Determine how many multiples of 60 are within the range of 200 to 500.
To find the number of multiples of 60 within a range, we need to find the largest multiple of 60 that is less than or equal to the upper bound of the range (500), and the smallest multiple of 60 that is greater than or equal to the lower bound of the range (200).
The largest multiple of 60 less than or equal to 500 is 480 (60 x 8), and the smallest multiple of 60 greater than or equal to 200 is 240 (60 x 4).
Step 3: Calculate the number of multiples.
To find the number of multiples within the range, we subtract the smallest multiple from the largest multiple and add 1 to include both endpoints:
Number of multiples = (largest multiple - smallest multiple) + 1
= (480 - 240) + 1
= 241
So, there are 241 positive integers between 200 and 500 that are divisible by 4, 6, 10, and 12.