Question

Find the sum of all two digit numbers which are either multiples of 2 or 3

Answer 1

two digit numbers are  : 10,11,12,....,,98,99

numbers divisible by 2 are  10, 12, 14, ...96, 98.   series AP
numbers divisible by 3 are  12, 15, 18,... , 96, 99
there are some common numbers in the two series: 12, 18, 24, ...,90, 96. 

A.P. Series1  =  10, 12, ..,96, 98
      number of terms : (98-10)/2 + 1 = 45
     S1 = (first term+last term)/2 * number of terms 
          = (10+98)/2*45 = 27*45

sum of numbers divisible by 3 :  
      number of terms = (99 - 12)/3 + 1  = 28
      S2 = (99+12) / 2 * 28  = 111 * 14

Sum of numbers divisible by  6 : 
       number of terms = (-12 + 96)/6 + 1 = 15
            S3 = (96+12)/2 * 15 = 54 * 15

Answer  =  S1 + S2 - S3