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M07-14 09 Aug 2023, 09:54
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I think this is a high-quality question and I agree with explanation.
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M07-14 18 Dec 2023, 06:38
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Hi Bunuel

What if the question is, Integers between 1 and 1000(inclusive) that are not divisible by 11 and 35?
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M07-14 18 Dec 2023, 06:54
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Rishitha0311
Hi Bunuel

What if the question is, Integers between 1 and 1000(inclusive) that are not divisible by 11 and 35?
Show more

I'd say it would be an ambiguous rewording of the original question. The ambiguity arises from the use of the phrase "not divisible by 11 and 35," which can be interpreted in different ways depending on how one understands the logical relationship implied by "and." This ambiguity makes it less clear compared to the original question, which more straightforwardly uses "either...or" to establish the logical relationship between the divisibility by 11 and 35.
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M07-14 18 Dec 2023, 07:44
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Bunuel
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Hi Bunuel

What if the question is, Integers between 1 and 1000(inclusive) that are not divisible by 11 and 35?

I'd say it would be an ambiguous rewording of the original question. The ambiguity arises from the use of the phrase "not divisible by 11 and 35," which can be interpreted in different ways depending on how one understands the logical relationship implied by "and." This ambiguity makes it less clear compared to the original question, which more straightforwardly uses "either...or" to establish the logical relationship between the divisibility by 11 and 35.
Show more


Thanks for the clarification, Bunuel. Please help in understanding 2 more points.
1. So are we saying that "not either A or B" and "neither A nor B" represent the same? Am I correct?
2. What can be a right question formation if they had to ask the all the numbers from 1 to 1000 expect common multiples of 11 and 35? I'm unable to frame the question.
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M07-14 18 Dec 2023, 08:17
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Rishitha0311
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Hi Bunuel

What if the question is, Integers between 1 and 1000(inclusive) that are not divisible by 11 and 35?

I'd say it would be an ambiguous rewording of the original question. The ambiguity arises from the use of the phrase "not divisible by 11 and 35," which can be interpreted in different ways depending on how one understands the logical relationship implied by "and." This ambiguity makes it less clear compared to the original question, which more straightforwardly uses "either...or" to establish the logical relationship between the divisibility by 11 and 35.


Thanks for the clarification, Bunuel. Please help in understanding 2 more points.
1. So are we saying that "not either A or B" and "neither A nor B" represent the same? Am I correct?
2. What can be a right question formation if they had to ask the all the numbers from 1 to 1000 expect common multiples of 11 and 35? I'm unable to frame the question.
Show more

1. Yes, you are correct. In logical terms, "not either A or B" and "neither A nor B" essentially represent the same concept. Both phrases are used to exclude cases where either A or B (or both) is true.

  • "Not either A or B" means that neither A nor B is true.
  • "Neither A nor B" directly states that neither A is true nor B is true.

In the context of our divisibility question, both phrases would be used to find numbers that are not divisible by 11 and also not divisible by 35.

2. I'd say "How many integers are there between 1 and 1000, inclusive, that are not divisible by both 11 and 35?".
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M07-14 18 Dec 2023, 08:34
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Bunuel
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Rishitha0311
Hi Bunuel

What if the question is, Integers between 1 and 1000(inclusive) that are not divisible by 11 and 35?

I'd say it would be an ambiguous rewording of the original question. The ambiguity arises from the use of the phrase "not divisible by 11 and 35," which can be interpreted in different ways depending on how one understands the logical relationship implied by "and." This ambiguity makes it less clear compared to the original question, which more straightforwardly uses "either...or" to establish the logical relationship between the divisibility by 11 and 35.


Thanks for the clarification, Bunuel. Please help in understanding 2 more points.
1. So are we saying that "not either A or B" and "neither A nor B" represent the same? Am I correct?
2. What can be a right question formation if they had to ask the all the numbers from 1 to 1000 expect common multiples of 11 and 35? I'm unable to frame the question.
Show more

1. Yes, you are correct. In logical terms, "not either A or B" and "neither A nor B" essentially represent the same concept. Both phrases are used to exclude cases where either A or B (or both) is true.

  • "Not either A or B" means that neither A nor B is true.
  • "Neither A nor B" directly states that neither A is true nor B is true.

In the context of our divisibility question, both phrases would be used to find numbers that are not divisible by 11 and also not divisible by 35.

2. I'd say "How many integers are there between 1 and 1000, inclusive, that are not divisible by both 11 and 35?".[/quote]


Noted. Thanks for your patience!
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M07-14 31 May 2024, 05:00
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AndrewN

Bunuel
What is the number of integers from 1 to 1000, inclusive that are not divisible by 11 or by 35?

A. 884
B. 890
C. 892
D. 910
E. 945
The 30-second method: You can use a little number sense and PS technique to crack this one in no time at all. Ask yourself, how many times will 11 fit into 100? 9 times. This will repeat for each set of 100, since the upper limit is set at 1000.

\(9*10=90\)

So far, then, we have 90 numbers to remove. Now, repeat the process for 35, but with a little fine-tuning. How many times will 35 fit into 100? 2 times. Into 200? 5 times, since the second hundred starts at 105. There cannot be more than three instances of 35 fitting into any given 100, since 3 * 35 = 105. For each set of 200, then, we should get about five 35s, and there are five sets of 200 in 1000. (We need not concern ourselves with the exact sequence of 2s and 3s per 200.)

\(5*5=25\)

There should be around 90 + 25, or 115 numbers to remove from consideration. Yes, there will be an overlap for each instance in which 11 and 35 cross paths, but even if 11 were 10, that would only happen twice out of our range of numbers.

\(1000-115=885\)

The answer must lie within 2 of 885, so (A), 884, is the only option that works. We can choose (A) with 100 percent confidence and spare ourselves the mental energy we may need for the next challenge.

- Andrew
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­I tried by doing this method but multipled 2*10 for the 35 instead of 5*5, by your method the answer comes out to be 885 but the actual answer is 884. What would happen if there was 885 in the given options then what should be done 
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M07-14 15 Nov 2024, 07:09
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Bunuel
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
Show more

Here's a quick way to solve this question:

Step 1: Calculate number of multiples for each value given (11 and 35).

→ Number of multiples of 11 between 1 and 1000 = ⌊1000 / 11⌋ = 90 terms - (1)

These brackets "⌊" and "⌋" are called floor brackets (or function); they give us the greatest integer that is less than or equal to a given number between these brackets.

→ Similarly, number of multiples of 35 between 1 and 1000 = ⌊1000 / 35⌋ = 28 terms - (2)

→ Total number of terms = 90 + 28 = 118 terms - (3)

Step 2: Calculate the number of common multiples for the values given (11 and 35).

→ There are some common terms as well; LCM (35,11) = 385. Multiples of 385 between 1 and 1000 = ⌊1000 / 385⌋ = 2 terms - (4)

Note that these terms are already counted in step 1.

Step 3: Net total number of terms between 1 and 1000 that are a multiple of 11 AND 35 = (3) - (4) = 118 - 2 = 116 terms.

Step 4: Therefore, the number of terms which ARE NOT multiples of 11 and 35 between 1 and 1000 are = 1000 - 116 = 884 terms, which is option A.
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M07-14 27 May 2025, 08:25
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I like the solution - it’s helpful.
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M07-14 13 Jun 2025, 07:59
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M07-14 18 Jun 2025, 16:21
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Bunuel for calculation number of multiples, can we not solve it using below theory text:

The number of multiples of an integer d less than or equal to N is = [N/d]

Bunuel
Quote:
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:

\(\frac{\text{last multiple in the range - first multiple in the range} }{\text{multiple} }+1\)

Thus:

• The number of multiples of 11 in the given range is \(\frac{ last - first}{multiple}+1=\frac{990-11}{11}+1=90\);

• The number of multiples of 35 in the given range is \(\frac{ last - first }{ multiple}+1=\frac{980-35}{35}+1=28\);

• The number of multiples of both 11 and 35 is 2 (since \(11*35=385\) and \(385*2=770\));

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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M07-14 17 Nov 2025, 03:34
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