How many integers between 100 and 500 inclusive are divisible

Question

How many integers between 100 and 500 inclusive are divisible by either 4 or 6?

(A) 102
(B) 124
(C) 132
(D) 135
(E) 148

(I'm confused - should either...or be exclusive or inclusive? Like A+B-2*A&B or A+B-A&B)

GMATinsight · Answer

Divisible by 4 from 1 to 500 =   = 125
Divisible by 4 from 1 to 100 =   = 25
Divisible by 4 from 100 to 500 inclusive = 125-25 +1 (to include 100) = 101

Divisible by 6 from 1 to 500 =   = 83
Divisible by 6 from 1 to 100 =   = 16
Divisible by 6 from 100 to 500 inclusive = 83 - 16 = 67

Divisible by 12 (LCM of 4 and 6) from 1 to 500 =   = 41
Divisible by 12 from 1 to 100 =   = 8
Divisible by 12 from 100 to 500 inclusive = 41 - 8 = 33

Total desirable numbers = 101 + 67 - 33 (common numbers) = 135

Answer: Option D

Percheron  GMAT has used this language always to be inclusive of numbers that are divisible by both divisors. Therefore, you need to change the answer of this question from Option A to Option D

Related Video:

https://www.youtube.com/watch?v=IuWiDWQruVQ

Kinshook · Answer

Percheron

How many integers between 100 and 500 inclusive are divisible by either 4 or 6?

Numbers divisible by 4 = {100,104,..... 500} = (500-100)/4 + 1 = 101
Numbers divisible by 6 = {102,108,......498} = (498-102)/6 + 1 = 67
Numbers divisible by both 4&6 or LCM(4,6)=12 = {108,.....492} = (492-108)/12 + 1 = 33

Numbers divisible by either 4 or 6 = 101 + 67 - 33 = 135 since 33 is counted twice

IMO D

himomaki · Answer

We need to calculate how many numbers between 100 and 500 are divisible by 4 or 6.

Numbers divisible by 4: The sequence starts at 100 and ends at 500. The formula (500−100)/4+1 gives 101 numbers divisible by 4.

Numbers divisible by 6: Starting at 102 and ending at 498, we get 67 numbers divisible by 6 using the formula (498−102)/6+1

Numbers divisible by both 4 and 6 (i.e., divisible by 12): The sequence starts at 108 and ends at 492. Using the formula (492−108)/12+1, we get 33 numbers.

By the principle of inclusion-exclusion, the total number of numbers divisible by 4 or 6 is: 101+67−33=135

Thus, the answer is 135.

Percheron · Answer

I asked ChatGPT - it says "GMAT-Style Tip:
In  logic or math  questions,  “either A or B”  often means  exactly one is true , not both.

Is there an official clarification for that?

Krunaal · Answer

In such questions, "either A or B" is generally interpreted inclusively unless the problem explicitly states "but not both", if the problem had specified “either 4 or 6,  but not both ,” then we would exclude numbers divisible by both 4 and 6 (i.e., those divisible by 12)

It also depends on the context, for example, "Either Sarah or Kim will be the team leader" - here the sentence suggests that either of the one is possible. So, there are no set official rules.

How many integers between 100 and 500 inclusive are divisible

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How many integers between 100 and 500 inclusive are divisible

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