How many numbers between 0 and 200 are divisible by both 6 and 8?

To figure this out, you don't need to test every number individually. The secret lies in finding the smallest number that both 6 and 8 can divide into evenly. This is called the Least Common Multiple, or LCM. The first number that appears in both the multiplication table of 6 (6, 12, 18, 24...) and 8 (8, 16, 24...) is 24. This means that any number divisible by both 6 and 8 must also be a multiple of 24.

With this shortcut, the problem becomes much simpler. We just need to find out how many multiples of 24 exist between 0 and 200. We can do this by dividing the upper limit of our range by our new "magic number": 200 divided by 24 equals 8.33. Since we are only looking for whole numbers that fit completely within the range, we disregard the decimal. The whole number, 8, is our count.

This mathematical concept isn't just for trivia; it's used frequently in the real world. The LCM is essential for adding or subtracting fractions with different denominators (like 1/6 + 1/8). It's also used in scheduling. If one bus arrives every 6 minutes and another arrives every 8 minutes, the LCM of 24 tells you they will both arrive at the same stop together every 24 minutes.