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

Please post explanation

agarwalva 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

Please post explanation

Strategy: First find the numbers that do not contain a 4 in this range and then subtract all the multiples of 4 from it.

How many numbers between 1 and 400 do not contain 4 as a digit?
one-digit numbers: 8
two digit numbers: 8C1*9C1 = 72
three-digit numbers: 3C1*9C1*9C1 = 243
Total = 8 + 72 + 243 = 323

Which numbers between 1 and 400 are multiples of 4 and do not contain 4?
Numbers ending in 08,12,16,20,28,.. etc

Strategy: Find all the multiples of 4 and then subtract those that contain a 4. These numbers will then be subtracted from 323 to get the required answer.

From 00 to 99, there are a total of 25 multiples of 4 but 7 of them (04, 24, 40, 44, 48, 64, 84) contain 4 in them.
So, there are (25 - 7 =) 18 numbers from 00 to 99 that do not contain a 4 but are multiples of 4.

There will be 18 numbers each between (00 - 99), (100 - 199).. (300 - 399)
So, 4*18 = 72 numbers in all.

Number of numbers from 1 to 400 that do not contain a 4 and are not multiples of 4
= 323 - 72
= 251

aneesh.kg wrote:

agarwalva 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

Please post explanation

Strategy: First find the numbers that do not contain a 4 in this range and then subtract all the multiples of 4 from it.

How many numbers between 1 and 400 do not contain 4 as a digit?
one-digit numbers: 8
two digit numbers: 8C1*9C1 = 72
three-digit numbers: 3C1*9C1*9C1 = 243
Total = 8 + 72 + 243 = 323

Which numbers between 1 and 400 are multiples of 4 and do not contain 4?
Numbers ending in 08,12,16,20,28,.. etc

Strategy: Find all the multiples of 4 and then subtract those that contain a 4. These numbers will then be subtracted from 323 to get the required answer.

From 00 to 99, there are a total of 25 multiples of 4 but 7 of them (04, 24, 40, 44, 48, 64, 84) contain 4 in them.
So, there are (25 - 7 =) 18 numbers from 00 to 99 that do not contain a 4 but are multiples of 4.

There will be 18 numbers each between (00 - 99), (100 - 199).. (300 - 399)
So, 4*18 = 72 numbers in all.

Number of numbers from 1 to 400 that do not contain a 4 and are not multiples of 4
= 323 - 72
= 251

Answer is 252 though!

agarwalva 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

Please post explanation

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
For an integer to be multiple of 4, its last two digits must form a multiple of 4.
Since 100/4 = 25, the number of two-digit multiples of 4 = 25.
But since the 324 options above do include a digit of 4, our options for the last two digits here must EXCLUDE any two-digit integer with a digit of 4.
Of the 25 multiples of 4 between 1 and 100, the following 7 options include a digit of 4: 04, 24, 40, 44, 48, 64, 84.
Thus, between 1 and 100, the number of multiples of 4 that DO NOT include a digit of 4 = 25-7 = 18.
Thus:
Number of options for the last two digits = 18.
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.

every forth number is divisible bye 4... 400/4= 100

--> now numbers contains 4 as digit
1) from 40-49 we have ten such numbers out of which tthree are divisible by 4 we have already considered above... so we have 7 such numbers in every hundered numbers, here till 400 we will have 7x4= 28 such numbers.

2)now we consider number whch contains 4 as digit but not divisiable by 4 (except Eq. 1)
14,34,54,74,94... comes once in every 20 numbers. so total such kind of numbers will be 400/20 = 20

* Resulting count is 400 - (100+28+20) = 252

GMATGuruNY wrote:

agarwalva 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

Please post explanation

Good = total - bad.

Total: Integers between 1 and 400 that do NOT include 4
In these sorts of problems:
A one-digit integer can be represented as 00X.
A two-digit integer can be represented as 0XX.
The result:
We need to count all of the integers 000 through 400 that do NOT include a digit of 4.

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
For an integer to be multiple of 4, its last two digits must form a multiple of 4.
Since 100/4 = 25, the number of two-digit multiples of 4 = 25.
But since the 324 options above do include a digit of 4, our options for the last two digits here must EXCLUDE any two-digit integer with a digit of 4.
Of the 25 multiples of 4 between 1 and 100, the following 7 options include a digit of 4: 04, 24, 40, 44, 48, 64, 84.
Thus, between 1 and 100, the number of multiples of 4 that DO NOT include a digit of 4 = 25-7 = 18.
Thus:
Number of options for the last two digits = 18.
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 first part is incorrect. "Integers between 1 and 400 that do NOT include 4" is not the same as "We need to count all of the integers 000 through 400 that do NOT include a digit of 4" because the latter include 0 which is not between 1 and 400. In fact, Integers between 1 and 400 that do NOT include 4 = your answer 324-1=323

BenjiP wrote:Integers between 1 and 400 that do NOT include 4" is not the same as "We need to count all of the integers 000 through 400 that do NOT include a digit of 4"

Correct.
To make the calculations easier, my solution above counts the number of integers between 0 and 399, inclusive, that do not include a digit of 4.
I've amended my post to make this clear.