What is the sum of all integers between 1 and 100 divisible

Arithmetic and Geometric progressions.

What is the sum of all integers between 1 and 100 divisible

Postby Guest » Mon Feb 11, 2013 4:09 am

What is the sum of all integers between 1 and 100 which are divisible by 2 or 5.





Guest
 

Re: What is the sum of all integers between 1 and 100 divisi

Postby Guest » Mon Feb 11, 2013 10:29 am

Use the inclusion exclusion principle.
The sum of all integers which are divisble by 2 or 5
= The sum of all integers which are disible by 2
+ The sum of all integers which are disvisible by 5
- The sum of all integers that are divisible by 2 and 5 (i.e. are divisible by 10)

= (2 + 4 + 6 + ... + 100) + (5 + 10 + 15 + ... 100) - (10 + 20 + 30 + ... + 100)
= 2(1+2+3+...+50) + 5(1+2+3+...20) -10(1+2+3+...+10)
= 2(50x51/2) + 5(20x21/2) -10(10x11/2) (using the fact that 1+2+3+...+n = n(n+1)/2)
= 3050

Guest
 

Re: What is the sum of all integers between 1 and 100 divisi

Postby Guest » Sun Oct 23, 2016 6:13 am

Read the question again.....

"What is the sum of all integers between 1 and 100 which are divisible by 2 or 5. "

It is the sum of two arithmetic progressions

1..............from 2 to 100 with common difference of 2

2..............from 5 to 100 with common difference of 5

That means all integers that are divisible by 2 OR 5.......not 2 AND 5

and the answer is .......3600

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Guest
 


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