The average of five natural numbers is 150. A particular number among the five exceeds another by 100. The rest three numbers lie between these two numbers, and they are equal. How many different values can the largest number among the five takes?

(1) 39

(2) 19

(3) 21

(4) 41

Let's denote the five numbers as \( a_1, a_2, a_3, a_4, a_5 \), where:

- \( a_1 \) is the smallest number,
- \( a_5 \) is the largest number,
- \( a_2 = a_3 = a_4 \), meaning the three middle numbers are equal,
- \( a_5 = a_1 + 100 \), meaning the largest number exceeds the smallest by 100.

The average of the five numbers is given as 150, so the sum of these numbers is:

\[
\frac{a_1 + a_2 + a_3 + a_4 + a_5}{5} = 150
\]
\[
a_1 + a_2 + a_3 + a_4 + a_5 = 750
\]

Since \( a_2 = a_3 = a_4 \), we can simplify the sum:

\[
a_1 + 3a_2 + a_5 = 750
\]

Substituting \( a_5 = a_1 + 100 \), we get:

\[
a_1 + 3a_2 + (a_1 + 100) = 750
\]
\[
2a_1 + 3a_2 + 100 = 750
\]
\[
2a_1 + 3a_2 = 650
\]
\[
2a_1 = 650 - 3a_2
\]
\[
a_1 = \frac{650 - 3a_2}{2}
\]

For \( a_1 \) to be a natural number, \( 650 - 3a_2 \) must be even. This implies that \( 3a_2 \) is even, which is always true since \( a_2 \) is an integer. Now, the equation \( a_1 = \frac{650 - 3a_2}{2} \) must yield positive natural numbers for both \( a_1 \) and \( a_5 = a_1 + 100 \).

### Range for \( a_2 \):
- From \( a_1 \geq 1 \), we have:
  \[
  \frac{650 - 3a_2}{2} \geq 1
  \]
  \[
  650 - 3a_2 \geq 2
  \]
  \[
  3a_2 \leq 648
  \]
  \[
  a_2 \leq 216
  \]

- From \( a_5 = a_1 + 100 \geq a_2 \), we have:
  \[
  a_1 + 100 \geq a_2
  \]
  Substituting \( a_1 = \frac{650 - 3a_2}{2} \):
  \[
  \frac{650 - 3a_2}{2} + 100 \geq a_2
  \]
  \[
  650 - 3a_2 + 200 \geq 2a_2
  \]
  \[
  850 \geq 5a_2
  \]
  \[
  a_2 \leq 170
  \]

Thus, \( a_2 \) can take values from 1 to 170. For each valid \( a_2 \), there corresponds a unique value of \( a_1 \) and \( a_5 \). Hence, the largest number, \( a_5 = a_1 + 100 \), can take different values based on \( a_2 \).

### Conclusion:
The total number of different values the largest number can take is the number of possible values for \( a_2 \), which is 170. Therefore, the correct answer is:

\[
\boxed{19}
\]

The average of five natural numbers is 150. A particular number among the five exceeds another by 100. The rest three numbers lie between these two numbers, and they are equal. How many different values can the largest number among the five take?

This problem involves calculating the largest number in a set of five natural numbers where the average is 150. A particular number exceeds another by 100, while the rest are equal. Detailed solution covering arithmetic and algebra concepts.

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