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

Question

A. 26
B. 29
C. 31
D. 32
E. 41

KarishmaB · Accepted Answer

The venn diagram doesn't give you the answer directly. You do have to do the calculations shown above.

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

Total = n(A) + n(B) + n(C) - n(A and B) - n(B and C) - n(A and C) + n(A and B and C) = 50 + 33 + 20 - 16 - 10 - 6 + 3 = 74

Number not divisible by 2, 3 or 5 = 100 - 74 = 26

Answer (A)

You can plug in the same numbers in a venn diagram and solve though using the formula will be more efficient here.

salaxzar · Answer

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.

PareshGmat · Answer

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

Numbers not divisible by 2, 3 & 5 = 100 - 74 = 26

Answer = A

Is the OA correct OR have I made some mistake?

PareshGmat · Answer

We can take inspiration from Venn diagram as shown below: ven.png

Raihanuddin · Answer

Have you got the answer using venn diagram?

Raihanuddin · Answer

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

PareshGmat · Answer

Its an illustration as pointed out by Karishma..... Karishma: Thanks for the correction; Kudos to you

zerosleep · Answer

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.

mvictor · Answer

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.

gracie · Answer

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

salaxzar · Answer

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!!

gradschool2021 · Answer

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

Fdambro294 · Answer

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  , inclusive

Now for the last Ten Numbers in the Range  , 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-

Posted from my mobile device

KarishmaB · Answer

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.

Ekland · Answer

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

reeyab · Answer

While looking at the options it becomes obvious that the question meant 2,3 "or" 5. However the "and" in the question is very misleading as it seemings like its asking about numbers which are multiples of 30, technically

Dteran17 · Answer

Can some explain how to get the amount of numbers divisible by each number please?

For example, how do you get that 16 number from 1 to 100 are divisible by 6?

Thanks!

Bunuel · Answer

The number of multiples of an integer within a range can be calculated using the following formula:
 
•  \frac{	ext{last multiple in the range - first multiple in the range} }{	ext{multiple} }+1 
 
Thus:
 
• The number of multiples of 6 in the given range is  \frac{ last - first}{multiple}+1=\frac{96 -6}{6}+1=16 .
 
Hope it helps.

A_Nishith · Answer

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

Total No. Divisible by (2,3,or 5) = 50 + 33 + 20 - 16 - 10 - 6 + 3 = 74

Number not divisible by (2, 3 or 5) = 100 - 74 = 26

Answer : A

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

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

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