How many integers are there between 1 and 100 which have 4 as a digit but are not divisible by 4 ?

The correct answer is Option 12 (or 12).

Step 1: Understand the requirements
We need to find the number of integers between 1 and 100 (inclusive) that satisfy two conditions:
1. They contain the digit '4'.
2. They are not divisible by 4.

Step 2: List all numbers between 1 and 100 that contain the digit '4'
Let's list all numbers from 1 to 100 having 4 in the units place or tens place:

• 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, 44, 45, 46, 47, 48, 49

Combining both lists without repeating 44, we get a total of 19 numbers:
4, 14, 24, 34, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 54, 64, 74, 84, 94

Step 3: Identify which of these numbers are divisible by 4
Now, let's check which of these 19 numbers are divisible by 4:

• 4 ÷ 4 = 1 (Divisible)
• 24 ÷ 4 = 6 (Divisible)
• 40 ÷ 4 = 10 (Divisible)
• 44 ÷ 4 = 11 (Divisible)
• 48 ÷ 4 = 12 (Divisible)
• 64 ÷ 4 = 16 (Divisible)
• 84 ÷ 4 = 21 (Divisible)

There are 7 numbers among them that are divisible by 4: {4, 24, 40, 44, 48, 64, 84}.

Step 4: Subtract the numbers divisible by 4
To find the numbers that have 4 as a digit but are not divisible by 4, we subtract the count of divisible numbers from the total count of numbers containing the digit 4:

$\text{Required Count}=19-7=12$

The numbers are: 14, 34, 41, 42, 43, 45, 46, 47, 49, 54, 74, 94 (a total of 12 integers).