All possible two-factor products are formed from the numbers$1, 2, ... , 100.$ The number of factors out of the total obtained which are multiple of 3, is 

The total number of two factor products$= 100C2.$ Out of the numbers$1, 2, 3, ... , 100$; the multiples of 3 are$3, 6, 9, ... , 99$i.e., there are 33 multiples of 3, and therefore there are 67 non-multiples of 3. 

So,    the number of two factor products which are not multiples of $3{=}^{67}{C}_2$

So, the required number  $=100\mathrm{C}2-67\mathrm{C}2=2739.$