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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
A. 884
B. 890
C. 892
D. 910
E. 945
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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
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 \(385*2=770\));
• 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
17 Jan 2012, 04:12
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manalq8
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?
