Problem Solving

How many of the integers between 1 and 400, inclusive, are not divisible by 4 and do not contain any 4s as a digit?

A. 72
B. 251
C. 252
D. 323
E. 324

The OA is C.

Please, can any expert explain this PS question for me? I have many difficulties to understand why that is the correct answer. Thanks.

LUANDATO wrote:How many of the integers between 1 and 400, inclusive, are not divisible by 4 and do not contain any 4s as a digit?

A. 72
B. 251
C. 252
D. 323
E. 324

Good = total - bad.

Total: Integers between 1 and 400 that do NOT include 4:
To make the calculations easier, count the THREE-DIGIT options between 000 and 399, inclusive, where 001 represents the single digit integer 1, 073 represents the 2-digit integer 73, and so on.
000 will be removed from the total when we subtract the BAD options -- the multiples of 4 -- since 0 is divisible by EVERY integer.

Number options for the hundreds place = 4. (0, 1, 2, or 3.)
Number of options for the tens place = 9. (Any digit but 4.)
Number of options for the units place = 9. (Any digit but 4.)
To combine these options, we multiply:
4*9*9 = 324.

Bad: Of the 324 options above, any integer that is a multiple of 4
From the 324 options above, we must subtract the multiples of 4.
An integer is a multiple of 4 if its last two digits form a multiple of 4.
Since 100/4 = 25, the number of two-digit multiples of 4 = 25.
Of these 25 2-digit multiples of 4, the following 7 options include a digit of 4: 04, 24, 40, 44, 48, 64, 84.
Thus, the number of 2-digit multiples of 4 that DO NOT include a digit of 4 = 25-7 = 18.
Thus -- of the 324 options above -- the multiples of 4 can be counted as follows:
Number of options for the last two digits = 18. (Since there are 18 2-digit integers that do not include a digit of 4).
Number of options for the hundreds digit = 4. (0, 1, 2, or 3.)
To combine these options, we multiply:
18*4 = 72.

Thus:
Good integers = total - bad = 324 - 72 = 252.

The correct answer is C.