How many integers are there between 1 and 1000, inclusive, that are

Official Solution:

How many integers are there between 1 and 1000, inclusive, that are not divisible by either 11 or 35?

A. 884
B. 890
C. 892
D. 910
E. 945


Let's determine the number of multiples of 11 or 35 between 1 and 1000, inclusive, and subtract that number from 1000.

The number of multiples of an integer within a range can be calculated using the following formula:

Thus:

Observe that the two numbers 385 and 770 are included in the count of multiples of 11 and the count of multiples of 35. To avoid double-counting, we need to subtract these numbers once from the total count of multiples of 11 and 35. Therefore, the number of multiples of either 11 or 35 in the given range is \(90+28-2=116\).

Consequently, the count of numbers that are not divisible by either 11 or 35 is \(1000-116=884\).


Answer: A­
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Normally, I would use the method used by Bunuel. It's the most accurate. But if you are looking for a speedy solution, you can use another method which will sometimes give you an estimate. Looking at the options (most of them are spread out), I wont mind trying it. (Mind you, the method is accurate here since the numbers start from 1.)

In 1000 consecutive numbers, number of multiples of 11 = 1000/11 = 90 (Ignore decimals)
In 1000 consecutive numbers, number of multiples of 35 = 1000/35 = 28
Number of multiples of 11*35 i.e. 385 = 1000/385 = 2

Number of integers from 1 to 1000 that are divisible by neither 11 nor by 35 = 1000 - (90 + 28 - 2) {Using the concept of sets here) = 884

Think: Why did I say the method is approximate in some cases?
Think what happens if the given range is 11 to 1010 both inclusive (again 1000 numbers)
What is the number of multiples in this case?
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Hello,

Calculate the no. of terms from 1 to 1000 (inclusive) that are divisible by 11 or 35 or both.

1. No of terms divisible by 11 -> 1000/11 = 90
2. No of terms divisible by 35 -> 1000/35 = 28
3. No of terms divisible by 11 and 35 -> 1000/(11*35) = 2

Answer = 1000- (90+28-2) = 884.

Number divisible by 11:-
1000/11= 90

numbers divisible by 35:-
1000/35= 28

Numbers divisible by both 11 and 35= 2

Total numbers divisible by both 11 and 28= 90+28-2= 116 (because we counted 2 in both 90 and 28)

Total numbers not divisible by 11 or 35= 1000-116= 884

A is the answer
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Numbers divisible by 11: 1001/11 = 91
Numbers divisible by 35: 1001/35 = 28
Numbers divisible by 11x35: 1001/385 = 2

1001-(91+28)+2 = 884

A.

­Can you explain why you check for numbers dividible by 11x35?

Thank

­
I updted the solution HERE. Hope now it's clear.
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­Hello Bunuel. Could you please help me understand a quick way to find out the greatest multiple of 35? How did you got within less than 2 minutes to 980?
It would help me a lot when using this method with big numbers like these.

­ruis You can divide 1000 by 35, and then subtract the remainder, 20, from 1000:



 ­
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Thats awesome! Thanks

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