Question

Find the
(i) sum of those integers between 1 and 500 which are multiples of 2 as well as of 5.
(ii) sum of those integers from 1 to 500 which are multiples of 2 as well as of 5.
(iii) sum of those integers from 1 to 500 which are multiples of 2 or 5.

Answer 1

Multiples of 2 from 1 to 500 are 2, 4, 6, ..., 500 (A.P.)
Here, a=2, d=4-2=6-4=2
Let `a_(n)=500`
`rArr 2+(n-1)2=500 rArr n=250`
and `S_(250)=(250)/(2)(2+500)=125xx502=62750`
Multiples of 5 from 1 to 500 are 5, 10, 15, ... (A.P)
Here, `a=5, d=10-5=15-10=5`
Let ` a_(n)=500`
`rArr 5+(n-1)5=500 rArr n=100`
and `S_(100)=(100)/(2)(5+500)=50xx505 = 25250`
L.C.M. of 2 and 5 is 10
`:.` Multiples of 10 from 1 to 500 are 10, 20, 30, ... 500 (A.P)
Here, `a=10, d=20-10=30-10=10`
Let `a_(n)=500`
`rArr 10(n-1)10=500 rArr n=50`
and `S_(50)=(50)/(2)(10+500)=25xx510=12750`
Now, the sum of all integers from 1 to 500 and multiples of 2 or 5 =sum of multiples of 2 + sum of multiples of 5 - sum of multiples of 10
` = 62750+25250-12750=75250`