There are 49 numbers divisible by 2 between 1 and 100
there are 33 numbers divisible by 3 between 1 and 100
there are 16 numbers divisible by 6 (common to multiples of 2 and 3)
Hence numbers not divisible by 2 and 3 = 100 - (49+33-16) = 34 IMO
number properties
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shankar.ashwin
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Shouldn't it be 98-(49+33-16)=32 ?
Anil Gandham
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The wording of this question needs to be clarified.sud21 wrote:How many numbers between 1 and 100 are not divisible by 2 and 3?
30 31 32 33 34
Not divisible by 2 AND 3 means NOT DIVISIBLE BY 6.
The intended meaning here seems to be not divisible by 2 OR 3.
The GMAT likely would say INTEGERS instead of numbers.
BETWEEN x and y generally means NOT INCLUDING x and y, but I suspect that the intended meaning here is between 1 and 100, INCLUSIVE.
If this question were to appear on the GMAT, the wording would make the intended meaning crystal clear.
The solution below presumes that the question intends to ask the following:
How many integers between 1 and 100, inclusive, are neither even nor a multiple of 3?
This is an overlapping groups problem.
The big idea is to SUBTRACT THE OVERLAP.
Total integers = Even integers + Multiples of 3 - Multiples of 6 + Integers neither even nor a multiple of 3.
When we count the even integers and the multiples of 3, the OVERLAP between the two groups -- the multiples of 6 -- is counted twice.
Hence, the overlap -- the multiples of 6 -- must be SUBTRACTED from the total, as shown in the equation above.
Total integers = 100.
Even integers = 50. (Half are even, half are odd.)
Multiples of 3 = 100/3 = 33. (Ignore the decimal. Since 3 can be divided into 100 at most 33 times, there are 33 multiples of 3 between 1 and 100, inclusive.)
Multiples of 6 = 100/6 = 16. (Ignore the decimal. Since 6 can be divided into 100 at most 16 times, there are 16 multiples of 6 between 1 and 100, inclusive.)
Plugging these values into the equation above:
100 = 50 + 33 - 16 + N
98 = 67 + N
N = 33.
The correct answer is D.
For a similar but trickier problem, check my post here:
https://www.beatthegmat.com/how-many-int ... 88553.html
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ArunangsuSahu
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1. Step 1:
Total no of numbers divisible by 2
100=2+(n-1)*2
n=50
2. Step 2:
NUmbers divisible by 3 =33
3. Numbers divisible by 6=16
Not Divisible by 2 and 3= 100-(50+33-16)..By Set theory
=33
Total no of numbers divisible by 2
100=2+(n-1)*2
n=50
2. Step 2:
NUmbers divisible by 3 =33
3. Numbers divisible by 6=16
Not Divisible by 2 and 3= 100-(50+33-16)..By Set theory
=33
