How many integers between 1 and 2014 are divisible by either

How many integers between 1 and 2014 are divisible by either 5, 7 or 11.

Solution

Number divisible by 5={5,10,...,2010}

we need to check how many numbers between 5 and 2010 are divisible by 5 which is given by
(2010-4)/5 = 2006/5. The nearest integer count greater than 2006/5 is 401.

Number divisible by 7={7,14,....2009}

we need to check how many numbers between 7 and 2009 are divisible by 7 which is given by
(2009-6)/7 = 2003/7. The nearest integer count greater than 2003/7 is 286.

Number divisible by 11={11,22,...,2013}

we need to check how many numbers between 11 and 2013 are divisible by 11 which is given by
(2013-10)/11 = 2003/11. The nearest integer count greater than 2003/11 is 182

Now we check how many numbers are divisible by 5 or 7

Now we need to eliminate the numbers which were divisible by both 5 and 7.
Numbers divisible by both - {35, ,....,1995}
This can be done by (1995-34)/35 . The nearest count greater than is 56.

Using Inclusion-Exclusion principle,
|A U B| = |A| + |B| - |A intersection B|

so number of elements are divisible by 5 or 7=401+286-186=701

also the number of elements divible by 7 and 11={77,..2002}

so nunber of elementsboth divible by 7 and 11=2006-76/77=25

so number of elements are divisible by 7 or 11=286+186-25=447

elements divisible by 11 and 5={55,...1815}

number of elements divisible by 5 and 11=1815-54/55=32

elements divible by 5,7,11={385,...1925}

number of elements divisible by 5,7,and 11=1925-384/385=4

so number of elements divisible by 5,7 or11=401+286+186-56-25-32-4=756

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