Question

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

Answer 1

Answer:

12750

step by step explanation

to find

the sum of those integer between 1 and 500 which are the multiple of 2 as well

as 5.

let to find the lcm of 2 and 5

that is lowest common multiple

lcm of 2 and 5 will 10

and,

here,  

last number = 500

that is also divisible by both 2 and 5

so,

last number = 500,

let that numbers are in AP

so

here,

1st term (a)  = 10

last term (an)  = 500

common difference(d) = 10

now,

we know that,

by the nth term formulae of AP,

an = a + (n - 1)d

putting the values,

500 = 10 + (n - 1)10

10(n - 1) + 10 = 500

10(n - 1) = 500 - 10

10(n - 1) = 490

n - 1 = 490/10

n - 1 = 49

n = 49 + 1

n = 50

so,

number of terms(n) = 50

now,

by the sum formulae,

1st method

Sn = n/2{2a + (n - 1)d}

putting the values,

Sn = 50/2{2(10) + (50 - 1)10}

Sn = 25{20 + 49(10)}

Sn = 25(20 + 490)

Sn = 25(510)

Sn = 12750

___________________

2nd method

Sn = n/2(a + l)

where,

l = last term,

now,

putting the values,

Sn = 50/2(10 + 500)

Sn = 25(510)

Sn = 12750

so,

the sum of the numbers between 1 and 500 which are the multiples of both 2 and 5

= 12750

Answer 2

Answer:

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 

Hope it Helps !!!!