For the integers from 1 to 1000, how many multiples of 9 are also multiples of 6?
To find how many multiples of 9 are also multiples of 6 among the integers from 1 to 1000, we can follow these steps:
1. Determine the multiples of 9:
We divide 1000 by 9 to find the largest whole number multiple of 9 that is less than or equal to 1000. In this case, 1000 ÷ 9 equals 111 with a remainder of 1. Therefore, the largest whole number multiple of 9 that is less than or equal to 1000 is 111. So, there are a total of 111 multiples of 9 in the range from 1 to 1000.
2. Determine the multiples of 6:
Similarly, we divide 1000 by 6 to find the largest whole number multiple of 6 that is less than or equal to 1000. 1000 ÷ 6 equals 166 with a remainder of 4. Therefore, the largest whole number multiple of 6 that is less than or equal to 1000 is 166. So, there are a total of 166 multiples of 6 in the range from 1 to 1000.
3. Determine the multiples of both 6 and 9:
To determine the multiples that both 6 and 9 share, we need to find the least common multiple(LCM) of 6 and 9, which is 18.
Now, we divide the largest whole number multiple of 9 (111) by 18: 111 ÷ 18 equals 6 with a remainder of 3. This means that out of the 111 multiples of 9, only 6 multiples are divisible by 18.
4. Final Answer:
Therefore, out of the integers from 1 to 1000, there are 6 multiples of 9 that are also multiples of 6.