SOLUTION: How many positive integers not exceeding 3278 are divisible by neither 12 nor 15?

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Question 370070: How many positive integers not exceeding 3278 are divisible by neither 12 nor 15?
Found 2 solutions by Fombitz, Edwin McCravy:
Answer by Fombitz(32388)   (Show Source): You can put this solution on YOUR website!
Look at the number of integers from 1 to 3278 divisible by 12.

Look at the number of integers from 1 to 3278 divisible by 15.

Look at the number of integers from 1 to 3278 divisible by both 12 and 15(multiples of 60).

Then find the number divisible by 12 and 15.


So then the remaining integers from 1 to 3278 are not divisble by 12 or 15.
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
There are 3278 positive integers not exceeding 3278, so we will find the
number of positive integers which are multiples of 12 or 15 that do not
exceed 3278 and subtract that number from 3278.

3278÷12 = 273.16666667 so there are 273 multiples of 12 not exceeding 3278
3278÷15 = 218.53333333 so there are 218 multiples of 15 not exceeding 3278

We use the formula:

N(A or B) = N(A) + N(B) - N(A and B) 


N(multiples of 12 OR multiples of 15) = 

N(multiples of 12) + N(multiples of 15) - N(multiples of 12 AND multiples of 15) 

For a positive integer to be a multiple of both 12 and 15 it must contain
all the factor of both 12 and 15.

12 = 2x2x3 and 15 = 3x5

So if an integer is a multiple of both 12 and 15 it has factors 2x2x3x5
and is therefore a multiple of 60, the LCM of 12 and 15.

3278÷60 = 54.63333333 so there are 54 multiples of 60 not exceeding 3278.

Therefore 

N(multiples of 12 OR multiples of 15) = 

N(multiples of 12) + N(multiples of 15) - N(multiples of 12 AND multiples of 15)
 
which equals

273 + 218 - 54 = 437.

Subtracting this from all 3278 positive integers that do not exceed 3278,

we get 3278 - 437 = 2841.

That's the answer, 2841.

Edwin
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