Given:
- a and b are positive integers
- a/b = 0.6

To find:
Which of the following CANNOT be the value of a?

Solution:
To solve this problem, we need to understand the concept of rational numbers and their decimal representations. A rational number is any number that can be expressed as a fraction, where the numerator and denominator are both integers.

The decimal representation of a rational number can be terminating (the decimal representation ends) or non-terminating and repeating (the decimal representation has a repeating pattern). For example, 0.6 is a terminating decimal because it ends after one decimal place.

To express a fraction as a decimal, we divide the numerator by the denominator. In this case, a/b = 0.6.

Step 1: Express 0.6 as a fraction
To express 0.6 as a fraction, we need to determine the denominator that will result in a terminating decimal. Since 0.6 has one decimal place, the denominator should be a power of 10. Therefore, we can rewrite 0.6 as 6/10 or simplify it to 3/5.

Step 2: Analyze the answer choices
Now, let's analyze each answer choice to determine if it can be the value of a.

a) 42
To check if 42 can be the value of a, we divide 42 by 5. The result is 8.4, which is not equal to 0.6. Therefore, a = 42 can be a valid value.

b) 105
To check if 105 can be the value of a, we divide 105 by 5. The result is 21, which is not equal to 0.6. Therefore, a = 105 can be a valid value.

c) 137
To check if 137 can be the value of a, we divide 137 by 5. The result is 27.4, which is not equal to 0.6. Therefore, a = 137 can be a valid value.

d) 174
To check if 174 can be the value of a, we divide 174 by 5. The result is 34.8, which is not equal to 0.6. Therefore, a = 174 can be a valid value.

Conclusion:
Out of the given answer choices, the only value that cannot be the value of a is 137. Therefore, the correct answer is option C.