How many of the integers between 1 and 400, inclusive, are not

Question

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

uchihaitachi · Accepted Answer

This is a great question. 
There are a total of 400 numbers. 
The numbers divisible by 4 are 100. so we remove these 100 numbers.

Now we need to remove the numbers that contain the digit 4. 
400 is already removed as it is divisible by 4. 
Now, we need to remove numbers of the type 
x4y
so the numbers 41,42,43,45,46,47 and 49. 44 and 48 are already removed as they are divisible by 4. 
there are a total of 7 numbers in 1-100 and a total of 28 numbers in 1-400.

now we need to numbers of the type xy4 and the numbers are 14,34,54,74 and 94. as 24,44,64,84 are already removed because they are divisible by 4. 
there are a total of 5 numbers of this type in 1-100 and a total of 20 numbers in 1-400.

The total numbers that need to be removed are 100+28+20 = 148

Remaining numbers = 400 - 148 = 252.

OA - C

Luckisnoexcuse · Answer

Since their are total 400 numbers..100 will be divided by 4 hence our answer will be less than 300 option D and E are out
now numbers which have digit 4 but not divisible by 4
14, 34, 41, 42, 43, 45, 46, 47, 49, 54, 74, 94 = 12 nos.
Since last two digits are not divisible by 4 we can safely put one more digit in the next numbers
114, 134, 141, 142, 143, 146, 147, 149, 154, 174, 194
214, 234, 241, 242, 243, 246, 247, 249, 254, 274, 294
314, 334, 341, 342, 343, 346, 347, 349, 354, 374, 394

300 - 48 = 252 nos.
Option C

pushpitkc · Answer

Out of the total 400 integers that exist between 1 and 400, 
there are 300 integers which are not divisible by 4,
Since we are asked how many integers do not contain a 4 as a digit as well.
We have numbers 14, 34, 41, 42, 43, 45, 46, 47, 49, 54, 74, 94 in the range 1-100
which aren't divisible by 4.

Similarly, we have these integers in the ranges 100-200,200-300 and 300-400.
So there are a total of 48 integers which have a 4 but aren't divisible by 4.
So, total of such integers is 300 - 48 =  252 (Option C)

KanishkM · Answer

Between 1 and 400, We have 100 multiples of 4

We can eliminate D and E, as the total numbers become 300.

Now lets look for a pattern in 1 - 100 range 
1-10 0
10-20 1
30-40 1
40-50 7
50-60 1
60-70 0
70-80 1
80-90 0
90-100 1

Number of times 4 will be in that range => 12

Total will be 12*4 = 48

300-48 = 252

C

Shobhit7 · Answer

Integers between 1 and 400 inclusive: 400

Integers Divisible by 4: (400-4)/4 + 1 = 100

Integers with Digit 4 in hundreds place: 1 (400)
Integers with Digit 4 in tens place: 40 (4 x 1 x 10: digits for hundred 0,1,2,3 / for tens 4 / for unit 0-9)
Integers with Digit 4 in units place: 36 (4 x 9 x 1: digits for hundred 0,1,2,3 / for tens 0-9 except 4 / for unit 4)

Total: 100+1+40+36 = 177

Correction for double count in 177 integers: 29
Between 0-99: 7 Integers (4,24,40,48,64,84)
Similarly, 7 Integers each between 100-199, 200-299, 300-399.
Finally, 1 integer 400
 
Integers not divisible by 4 and do not contain 4: 400-(177-29) = 252. 
Ans C

oharsha12 · Answer

Can we solve this question by using factorial? Then subtract numbers that have 4s with 400? 
400/4 + 400/16 + 400/64 + 400/256

Sardor2512 · Answer

Bunuel  ,

Why don't we count 80, 84, 88 etc.?

Bunuel · Answer

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

All three 80, 84, and 88 ARE divisible by 4. Plus 84 contains 4 as its digit.

Archit3110 · Answer

total digits 400
and integers divisible by 4 from 1 to 400 ; 100 so left with 300 digits which are not divisible by 4
now digits which do not contain 4 and are not divisible by 4 ; 14, 34, 41, 42, 43, 45, 46, 47, 49, 54, 74, 94 we will have same no with hundereds as 1,2,3 so total 12*4 ; 48
answer 300-48 ; 252
IMO C

YashKothari94 · Answer

I took a slightly more conceptual approach, but I got the correct ans:

The Hundreds digit can take any value from 0 to 3 and thus there are 4 cases
The Tens digit can take 9 values (0-9 inclusive - '4')
Similarly the Units digit can take 9 values

Therefore total cases where none of the digits between 0-400 have a 4 = 4*9*9 = 324

Now, they should not be divisible by 4 as well. Therefore, the tens and units digit cannot be:
00, 08, 12,16,20,28,32,36,52,56,60,68,72,76,80,88,92,96. These are a total of 18 numbers. Notice that we have not included the multiples which have a 4 as a digit since that has been taken care of.

Now these 18 combinations can be with either 0,1,2 OR 3 as the hundreds digit. Therefore total cases - 18*4 = 72

Final ans is 324-72 = 252

nhiluu · Answer

No of integers divisible by 4 = (400-4)/4 + 1 = 100
No of integers divisible by 4 having "digit 4" = 20 + 8 + 1 = 29
ab4: Noted that the list of multiples of 4 has unit-digit pattern 4-8-2-6-0 every 5 integers (4, 8, 12, 16, 20, 24, 26, 28, 32, 36, 40, 44, 48, 52, 60...) --> 100/5 = 20 integers 
a4b: rule out 44, we have a40, a48 --> 2*4 = 8 integers
400: 1 integer
--> No of integers divisible by 4 not having "digit 4": 100 - 29 = 71
No of integers having "digit 4" = 4*9*9 - 1 = 323 integers (excluding 0)
--> No of integers not divisible by 4 and not having any 4s digit = 323 - 71 = 252 
C.

Adarsh_24 · Answer

Great question.

are not divisible by 4 and do not contain any 4s as a digit?

So if we are finding opposite and removing from total. Then the opposite should be (are divisible by 4) or (contain 4).

For some reason, this is not coming intuitively for me sometimes. Have to really think hard.

If you are getting confused like me at times, we can use one of De Morgan's law.
(A'.B')' = A+B

One more thing I will add, since 100 is perfectly divided by 4. The observations in first 100 will be very symmetric in next 100s too.
Eg: if 14 is not divisible by 4. then 114,214,314 will also be not divisible.
Similarly 24 divisible by 4. So 124,224,.. etc divisible by 4.

Akshaynandurkar · Answer

1 to 400 have 100 numbers that are divisible by 4
now check for numbers containing 4 as a digit but are not divisible by 4 (since we don't want the repetition)
for 1 to 9, no. of 4 - 0
10 to 20, no. of 4 - 1
21 to 30, no. of 4 - 0
31 to 40, no. of 4 - 2
41 to 50, no. of 4 - 6(41,43,45,46,47,49) ---be careful here
51 to 60, no. of 4 - 1
61 to 70, no. of 4 - 0
71 to 80, no. of 4 - 1
81 to 90, no. of 4 - 0
91 to 99, no. of 4 - 1

similarly for 100 to 199 , 200 to 299 and 300 to 400
we get total numbers with 4 but not divisible by 4 = 12 x 4 = 48

Total = 400-100-48 = 252

Schloop · Answer

Okay the other answers here are way too involved to answer in less than 2 minutes. Here's the easiest

Observation 1 - 400 contains a 4, making it a questions about the numbers 1-399

Observation 2 - 100 is divisible by 4, so the answer can be seen as 3 time the number of numbers meeting the criteria less than between 1-99.

Conclusion, the number must be divisible by 3

Observation 3, there are approx 30 numbers divisible by 4 between 1 and 100.

Observation 4, there are 100 numbers divisible by 4 between 1-400. 
Conclusion, number must be divisible by 3 and less than 300.

252 is the only valid option remaining

How many of the integers between 1 and 400, inclusive, are not

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How many of the integers between 1 and 400, inclusive, are not

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