Question

The sum of all the elements in the set {n ∈ {1, 2, ….., 100} | H.C.F. of n and 2040 is 1} is equal to _______.

Answer 1

∵ 2040 = 2³ x 3 x 5 x 17

Let A = Sum of all numbers which are divisible by 2 upto 100.

B = Sum of all numbers which are divisible by 3 upto 100

C = Sum of all numbers which are divisible by 5 upto 100

D = Sum of all numbers which are divisible by 17 upto 100

A ∪ B ∪ C ∪ D = (A + B + C + D) − (A∩B + A∩C + A∩D + B∩C + B∩D + C∩D) + (A∩B∩C + A∩B∩D + A∩C∩D + B∩C∩D) − (A∩B∩C∩D)

= (50 × 51 + 33 × 51 + 1050 + 51 × 5) − (51 × 16 + 550 + 102 + 315 + 51 + 85) + (180 + 0 + 0 + 0) − 0

= 3799

Required sum = 5050 − 3799 = 1251

Answer 2

2040 = 2³ × 3 × 5 × 17

n should not be multiple of 2, 3, 5 and 17.

Sum of all n = (1 + 3 + 5 ….. + 99) – (3 + 9 + 15 + 21 + ….. + 99) – (5 + 25 + 35 + 55 + 65 + 85 + 95) – (17)

= 2500 – 17/2 (3+99) – 365 –17

= 2500 – 867 – 365 –17

= 1251