Correct Option (C)

To determine the number of integers between 1 and 100 that contain the digit 4 but are not divisible by 4, a systematic enumeration and filtering process is applied:

Identify all integers between 1 and 100 (inclusive of 1 and 99) that contain the digit 4. These numbers are:
- Numbers with 4 in the units place: 4, 14, 24, 34, 44, 54, 64, 74, 84, 94.
- Numbers with 4 in the tens place: 40, 41, 42, 43, 45, 46, 47, 48, 49 (excluding 44, which is already listed).
The complete set of numbers containing the digit 4 is: {4, 14, 24, 34, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 54, 64, 74, 84, 94}. There are 19 such numbers.
From this identified set, exclude numbers that are divisible by 4. A number is divisible by 4 if the number formed by its last two digits is divisible by 4. The numbers from the list above that are divisible by 4 are:
- 4 (4 ÷ 4 = 1)
- 24 (24 ÷ 4 = 6)
- 40 (40 ÷ 4 = 10)
- 44 (44 ÷ 4 = 11)
- 48 (48 ÷ 4 = 12)
- 64 (64 ÷ 4 = 16)
- 84 (84 ÷ 4 = 21)
There are 7 such numbers.

Subtracting the numbers divisible by 4 from the total numbers containing the digit 4 yields the desired count: 19 - 7 = 12.

The specific numbers that satisfy both conditions (contain digit 4 but are not divisible by 4) are: 14, 34, 41, 42, 43, 45, 46, 47, 49, 54, 74, and 94.

Incorrect Options:

Options A (5), B (11), and D (13) are incorrect as they do not correspond to the accurate count derived from systematically identifying integers between 1 and 100 that contain the digit 4 and subsequently excluding those divisible by 4. The precise calculation, as detailed above, results in exactly 12 such integers.