how many integers, x, for which x/2, x/3 and x/5 is also an integer can be found between 1 and 500? help me!!

To solve this problem, we need to find the number of integers between 1 and 500 for which x/2, x/3, and x/5 are all integers.

We can break this down into two cases:

Case 1: x is divisible by both 2 and 3.
In this case, we need to find the integers that are divisible by both 2 and 3, which means they are divisible by their least common multiple (LCM), which is 6. We can calculate the number of integers satisfying this condition by counting the multiples of 6 between 1 and 500.

Number of multiples of 6 between 1 and 500:
(500 ÷ 6) - (1 ÷ 6) + 1 = 83 - 0 + 1 = 84

Case 2: x is divisible by both 2 and 5.
In this case, we need to find the integers that are divisible by both 2 and 5, which means they are divisible by their LCM, which is 10. We can calculate the number of integers satisfying this condition by counting the multiples of 10 between 1 and 500.

Number of multiples of 10 between 1 and 500:
(500 ÷ 10) - (1 ÷ 10) + 1 = 50 - 0 + 1 = 51

However, we have counted the integers that satisfy both Case 1 and Case 2 twice. So, we need to subtract the number of integers that are divisible by the LCM of 6 and 10, which is 30, in order to avoid the double counting.

Number of multiples of 30 between 1 and 500:
(500 ÷ 30) - (1 ÷ 30) + 1 = 16 - 0 + 1 = 17

Now, we can find the total number of integers satisfying the given condition by adding the results from both cases:

Total number of integers between 1 and 500 satisfying the condition = 84 + 51 - 17 = 118

Therefore, there are 118 integers, x, between 1 and 500 for which x/2, x/3, and x/5 are all integers.