Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

To find the natural numbers between 101 and 999 that are divisible by both 2 and 5, we first determine the least common multiple (LCM) of these two numbers. Since 2 and 5 are prime, their LCM is:

Calculating the LCM

The LCM of 2 and 5 is:

• LCM(2, 5) = 10

Identifying the Range

Next, we need to find the multiples of 10 within the range of 101 to 999.

Finding the First and Last Multiples

The smallest multiple of 10 greater than 101 is:

• 110

The largest multiple of 10 less than 999 is:

• 990

Counting the Multiples

Now, we can find how many multiples of 10 exist from 110 to 990. This can be calculated using the formula for the number of terms in an arithmetic sequence:

• First term (a) = 110
• Last term (l) = 990
• Common difference (d) = 10

The number of terms (n) can be calculated as:

n = ((l - a) / d) + 1

Substituting the values:

n = ((990 - 110) / 10) + 1 = (880 / 10) + 1 = 88 + 1 = 89

Final Count

Thus, the total number of natural numbers between 101 and 999 that are divisible by both 2 and 5 is:

89