How many integers from 1 to 100 are not divisible by 2, 3 and 5?

We can take inspiration from Venn diagram as shown below:

Have you got the answer using venn diagram?

yes, Karishma is correct.
By using formula we can get the answer easily.

Total = A+B+C -(sum of 2 overlap groups)+(all the three)+Neither

Its an illustration as pointed out by Karishma.....

Karishma: Thanks for the correction; Kudos to you

Hi,

We may also consider that - from 1 to 100 -> 1/2 of the numbers will be divided by 2 and 1/2 will not.
Similarly, from 1-100 -> 1/3 of the numbers will be divided by 3 and 2/3 will not.
And, from 1-100 -> 1/5 of the numbers will be divided by 5 and 4/5 will not.

Hence, No of number which are not divisible by 2,3 and 5 = 100(1/2)(2/3)(4/5) = 80/3 = 26.666. Ignoring the decimal since no of numbers cant be decimal leaves us with 26.

from 1 to 100, 50 numbers are even, and 50 are odd. therefore, we can eliminate right away 50 numbers.
now...multiples of 5...let's see the multiples of 5 that are not odd...
5, 15, 25, 35, 45, 55, 65, 75, 85, 95 - 10 numbers - another 10 numbers eliminated
multiples of 3 that are not multiples of 5 and are not odd...
3, 9, 21, 27, 33, 39, 51, 57, 63, 69, 81, 87, 93, 99 - another 14 eliminated.
50 - 10 - 14 = 26
answer is A.

Such questions are easier to crack if you understand one simple logic -

From 1 to 100 -> 1/n of the numbers will be divisible by n, and (1-1/n) will not - provided n is a prime no.
For example, if we consider, 3(prime) then from 1-100 :
1/3 of the numbers will be divided by 3 and (1-1/3 =2/3) will not be.
Similarly, 1/5 of the numbers will be divided by 5 and 4/5 will not be.

Now this question is a cakewalk -
No of number which are not divisible by 2,3 and 5 = 100*(1/2)*(2/3)*(4/5) = 80/3 = 26.666.
Since, we cant have decimal, ans should be 26.

can someone help me with my logic here?
If the number of integers from 1 to 100 that ARE divisible by 2, 3 and 5=3, (30, 60 and 90),
then the number of integers NOT divisible by 2, 3 and 5=100-3=97

Hi Gracie,
the questions really asks for all numbers that are either divisible by 2, 3 or 5.
This means, 2, 4, 6, 8, ....so on
3, 6, 9, 12, .....
5, 10, 15, 20, ....

Of course LCM (eg- 6, 15, 30, etc ) will be double counted so we must be careful while doing such questions!!

VeritasKarishma can you help me understand why the group of three is subtracted only once? I thought it was double counted and therefore needed to be subtracted twice. Could you explain when to apply each formula?

Numbers divisible by 2 = 50

Numbers divisible by 3 = 33

Numbers divisible by 5 = 20

Numbers divisible by 6 (LCM of 2 & 3) = 16

Numbers divisible by 10 (LCM of 2 & 5) = 10

Numbers divisible by 15 (LCM of 3 & 5) = 6

Numbers divisible by 30 (LCM of 2, 3 & 5) = 3

Numbers divisible by 2, 3 & 5 combined = 50 + 33 + 20 - (16 + 10 + 6) + 3 = 103 - 32 + 3 = 106 - 32 = 74

Quickest method possible:


(1st) using the same logic that underlies finding a Euler Number, we’ll first find the Euler Number of 90

1st, the Prime Factorization of 90 = 2 * (3)^2 * 5

Finding the Euler Number of 90 answers the following question: how many (+)Positive Integers before 90 are Co-Prime to 90?


(Meaning, how many numbers before 90 do NOT have a Prime Factor of: 2 , 3 , and 5)


——> 90 * (1 - (1/2)) * (1 - (1/3)) * (1 - (1/5) =

90 * (1/2) * (2/3) * (4/5) =

24 Numbers are NOT Multiples of 2, 3, OR 5 in the Integer Set [1 thru 90], inclusive


Now for the last Ten Numbers in the Range [91 - 100], inclusive:

Only TWO Numbers: 91 and 97 : fit the criteria. The rest are Multiples of 2 , 3 , or 5


There are 24 + 2 = 26 Numbers

-A-

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Numbers lying in two sets are subtracted. When you do that, the numbers lying in three sets are subtracted three times. So you have not counted them at all. Hence, you need to ADD back the numbers lying in all three sets.

Check here: https://anaprep.com/sets-statistics-thr ... ping-sets/

The area g is subtracted 3 times - once with each subtraction of 2 overlapping sets.
So you add g back at the end.

This question subtly tests your knowledge of prime factors; which is very basic in GMAT prep. You must be able to reel off the first 50 primes with closed eyes, otherwise you need to still focus on multiplication and division of numbers in your prep. No kidding!
Integers 1 thru 100 not divisible 2, 3, 5 include 1,7, 11,13, 17,19,23,29,31,37,41,43,47.....
So you could see for half the list it's 13 numbers. the other half is defnitely 13 as well.
13(2) = 26
Ans A

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