Question

find the sum of those integer between 1 and 500 which are multiples of 2 and as well as of 5.

Answer 1

Hello,
 We first find the LCM of 2 and 5 which is 10.
Now all those integers which are multiples of 10 are also the multiples of 2 and 5.
Therefore, multiples of 2 as well as of 5 between 1 and 500 are:
10, 20, 30, ...., 490 
Series forms an AP with first term,a=10 and common difference,d=20-10=10.
Let total number of terms of this AP be n.
Therefore, nth term of AP, an = Last term, l= 490 

an=a(n-1)d=l;
10+(n-1)10=490;
(n-1)10=480;
n-1=48;
n=48+1=49;
n=49

Thus,sum of n terms of AP is given as:
S₄₉=49/2 (10+490);
      =49/2 (500);
      =49×250=12250

bye :-)

Answer 2

The numbers from 1 to 500 which are multiple of 2 are:

2, 4, 6, 8.............500

This forms an AP, where

first term a = 2,

common difference = 4 - 2 = 2

last term l = 500

Toral numbers n = 500/2 = 250

Now, Sum = (n/2)*(a + l)

                = (250/2)*(2 + 500)

                = (250/2) * 502

                = 125 * 502

                = 62750

The numbers from 1 to 500 which are multiple of 5 are:

5, 10, 15, 20.............500

This forms an AP, where

first term a = 5,

common difference = 10 - 5 = 5

last term l = 500

Toral numbers n = 500/5 = 100

Now, Sum = (n/2)*(a + l)

                = (100/2)*(5 + 500)

                = 50 * 505

                = 25250

Again, multiple of 10 are included in both i.e. in multiple of 2 and multiple of 5 also.

Now, the numbers from 1 to 500 which are multiple of 10 are:

10, 20, 30.............500

This forms an AP, where

first term a = 10,

common difference = 20 - 10 = 10

last term l = 500

Toral numbers n = 500/10 = 50

Now, Sum = (n/2)*(a + l)

                = (50/2)*(10 + 500)

                = 25 * 510

                = 12750

 Now, the sum of integers from 1 to 500 which are multiple of 2 or 5 = sum of multiple of 2 + sum of multiple of 5 - sum of multiple of 2 and 5

                                                                                                       = 62750 + 25250 - 12750

                                                                                                       = 88000 - 12750

                                                                                                       = 75250