How many integers

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How many integers

by finance » Thu Aug 04, 2011 2:55 am
How many integers between 100 and 150, inclusive, cannot be evenly divided by 3 nor 5?

35
27
25
26
28

Is there any shortcut...what if the range would have been 100 and 1250?

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by Anurag@Gurome » Thu Aug 04, 2011 3:22 am
finance wrote:How many integers between 100 and 150, inclusive, cannot be evenly divided by 3 nor 5?

35
27
25
26
28

Is there any shortcut...what if the range would have been 100 and 1250?
Total no. of integers between 100 and 150, inclusive = 150 - 100 + 1 = 51
Out of these 51 integers, every 3rd integer would be divisible by 3, which means 1/3 rd of the total no. of integers would be divisible by 3, = (1/3) * 51 = 17
No. of integers between 100 and 150 divisible by 5 = 50/5 = 10, but here since 100 and 150 are both divisible by 5, so total no. of integers divisible by 5 = 11
No. of integers divisible by 5 * 3 (= 15) = 4 (integers are 105, 120, 135, and 150), but remember here that these 4 integers are already covered in the above 2 categories, so subtract these 4 integers from the final result.

Therefore, no. of integers divisible by 3 or 5 = 17 + 11 - 4 = 24
Hence, no. of integers not divisible by 3 or 5 = 51 - 24 = 27

The correct answer is B.
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by GMATGuruNY » Thu Aug 04, 2011 3:24 am
finance wrote:How many integers between 100 and 150, inclusive, cannot be evenly divided by 3 nor 5?

35
27
25
26
28

Is there any shortcut...what if the range would have been 100 and 1250?
If the range of integers were greater, I would treat this as an overlapping groups problem.
The big idea is to subtract the overlap.

Total integers = Multiples of 3 + Multiples of 5 - Multiples of 15 + Integers not multiples of 3 or 5

When we count the multiples of 3 and the multiples of 5, the overlap between the two groups -- the multiples of 15 -- will be counted twice.
Hence, the overlap -- the multiples of 15 -- must be subtracted from the total, as shown in the equation above.

To count evenly spaced integers:

Number of integers = (Biggest-Smallest)/Interval + 1.

When we count consecutive integers, the interval = 1.
When we count multiples of 3, the interval = 3.
When we count multiples of 5, the interval = 5.
When we count multiples of 15, the interval = 15.

Total integers from 100 to 150, inclusive:
(150-100)/1 + 1 = 51.

Multiples of 3 between 102 and 150, inclusive:
(150-102)/3 + 1 = 17.

Multiples of 5 between 100 and 150, inclusive:
(150-100)/5 + 1 = 11.

Multiples of 15 between 105 and 150, inclusive:
(150-105)/15 + 1 = 4.

Plugging these values into the equation above:

51 = 17 + 11 - 4 + N
N = 27.

The correct answer is B.
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by leumas » Sun Sep 18, 2011 4:58 am
finance wrote:How many integers between 100 and 150, inclusive, cannot be evenly divided by 3 nor 5?

35
27
25
26
28

Is there any shortcut...what if the range would have been 100 and 1250?
For 100 and 1250

No. of numbers between 102-1248 divisible by 3 = 383
No of integers between 100-1250 divisible by 5 = 231
No of integers between 105 - 1245 divisible by 15 = 77

I'm quoting Mitch: Total integers = Multiples of 3 + Multiples of 5 - Multiples of 15 + Integers not multiples of 3 or 5

Total integers between 100-1250 = 576

576=383+231-77+N
N= 49.