What is the average of all integers between 150 and 300 that are exactly divisible by 13?

The correct option is 227.5.

To find the average of all integers between 150 and 300 that are divisible by 13, we first need to identify the smallest and largest such integers in this range.

Step 1: Find the first term (smallest integer > 150 divisible by 13)
When we divide 150 by 13:
150 ÷ 13 = 11 with a remainder of 7.
To get the next multiple of 13 above 150, we add (13 - 7 = 6) to 150:
First term, a = 150 + 6 = 156.
(Verification: 156 ÷ 13 = 12)

Step 2: Find the last term (largest integer < 300 divisible by 13)
When we divide 300 by 13:
300 ÷ 13 = 23 with a remainder of 1.
Subtracting the remainder 1 from 300 gives the largest multiple of 13 below 300:
Last term, l = 300 - 1 = 299.
(Verification: 299 ÷ 13 = 23)

Step 3: Calculate the average
Since these numbers form an Arithmetic Progression (AP) with a constant common difference of 13, the average of all terms in the series is simply the arithmetic mean of the first term and the last term.

$\text{Average}=\frac{\text{First term}+\text{Last term}}{2}$

Substitute the values of the first term (156) and the last term (299):

$\text{Average}=\frac{156+299}{2}$

$\text{Average}=\frac{455}{2}=227.5$

Thus, the average of all integers between 150 and 300 that are exactly divisible by 13 is 227.5.