How many integers

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.

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.