Counting Multiples of 12 Between 200 and 290

The problem of counting how many multiples of 12 exist between 200 and 290 is a common exercise in basic arithmetic and number theory. There are several methods to solve this problem, each offering a unique perspective on the underlying principles of number ranges and multiples.

Method 1: Direct Multiplication

One approach involves listing out the multiples of 12 within the specified range. By multiplying 12 by consecutive integers, we can find the multiples of 12:

17 x 12 204 18 x 12 216 19 x 12 228 20 x 12 240 21 x 12 252 22 x 12 264 23 x 12 276 24 x 12 288

From this list, we can see the multiples of 12 that fall within the range of 200 to 290 are 204, 216, 228, 240, 252, 264, and 276, 288. Therefore, there are 8 multiples of 12 between 200 and 290.

Method 2: Range Calculation

Another effective method involves using the difference between the specified numbers. By first identifying the nearest integers to 200 and 290 when divided by 12, we can determine the multiples of 12:

200 / 12 16.667, so the nearest integer (ceiling) is 17. 290 / 12 24.167, so the nearest integer (floor) is 24.

Using these values, we calculate the total number of multiples of 12 between 200 and 290 as 24 - 17 7.

Next, we adjust by adding one to the final result because we are including both endpoints in the range.

Method 3: Subtract and Adjust

A simpler way to determine the count of multiples involves the following steps:

Divide 200 by 12, resulting in 16.667. The nearest integer (ceiling) is 17. Divide 290 by 12, resulting in 24.167. The nearest integer (floor) is 24. Subtract the floor value from the ceiling value, giving us 24 - 17 7. Add 1 to the result to include the first multiple after 200 and the last multiple before 290.

Thus, the total count is 7 1 8, and the multiples of 12 are 204, 216, 228, 240, 252, 264, and 276, 288.

In conclusion, whether you list the multiples directly, use the difference method, or perform a range calculation, the number of multiples of 12 between 200 and 290 is consistently 8 when including both endpoints.