Question

find the sum of all integers from 1 to 200 which are not multiples of 3 or 7.

Answer 1

The sum of all integers from 1 to 200 which are not multiples of 3 or 7
step 1
the sum of all integers from 1 to 200 = n x (n+1) / 2
                                                          = 200 x 201 /2
                                                          = 20100
step 2
the sum of all integers which are multiples of 3 between 1 to 200
   = 3+6+9+12+..................+198
   = 3 (1+2+3+4+................+ 66)
   =  3 (66 x 67)/2
   =  3 x 2211 = 6633
step 3
the sum of all integers which are multiples of 7 between 1 and 200
   = 7+14+21+28+...............+196
   = 7(1+2+3+4+...................+28)
   = 7(28 x 29)/2
   = 7 x 406 = 2842
the multiples of 3 and 7 = 6633 + 2842 =9475
 
step 4
so, solution = 20100 - 9475 = 10625