Question

Find the sum of those integers from 1 to 500 which are multiples of 2 or 5.

Answer 1

Since, multiples of 2 or 5 = Multiples of 2 + Multiples of 5 – [Multiples of LCM (2,5) i.e., 10],

∴ Multiples of 2 or 5 from 1 to 500

= List of multiples of 2 from 1 to 500 + List of multiples of 5 from 1 to 500 – List of multiples of 10 from 1 to 500

= (2, 4, 6,…, 500) + (5, 10,15,…, 500) – (10, 20,…, 500)  .......(i)

All of these list form an AP.

Now, number of terms in first list,

500 = 2 + (n₁ – 1)2

⇒ 498 = (n₁ – 1)2  .......[∵ a₁ = a + (n – 1)d]

⇒ n₁ – 1 = 249

⇒ n₁ = 250

Number of terms in second list,

500 = 5 + (n₂ – 1)5

⇒ 495 = (n₂ – 1)5  ......[∵ l = 500]

⇒ 99 = (n₂ – 1)

⇒ n₂ = 100

And number of terms in third list,

500 = 10 + (n₃ – 1)10

⇒ 490 = (n₃ – 1)10

⇒ n₃ – 1 = 49

⇒ n₃ = 50

From Eq(i) Sum of multiples of 2 or 5 from 1 to 500

= Sum of (2, 4, 6, …500) + Sum of (5,10,…, 500) – Sum of (10, 20, …, 500)

= `n_1/2[2 + 500] + n_2/2 [5 + 500] - n_3/2[10 + 500]`   .....`[because S_n = n/2(a + 1)]`

= `(250/2 xx 502) + (100/2 xx 505) - (50/2 xx 510)`

= (250 × 251) + (505 × 50) – (25 × 510)

= 62750 + 25250 – 12750

= 88000 – 12750

= 75250