This was a slightly difficult problem for many people, though can be solved with some persistent trial and error. Girish Tutakne, Pradeep Hengavelli, Suman Saraf, Chandu Karan and Pooja Goyal+Arushi Goyal were the ones to send correct answers.

I do not know if there is an elegant solution to this, requires trial and error. The solution is

Puzzle 119 Answer

The only trick one can think of is that the last digit of the two numbers cannot be “2” as the  multiplication will yield an even number as the last digit. Similarly, the last digits cannot be 3,3 or 7,7 or 7,3 as they yield 9, 9 and 1 respectively as the last digit. Therefore the combinations can be 5,5; 7,5 and 3,5. Beginning with these assumptions, one can build upon the different possibilities and rule them out gradually.

Hope you all enjoyed the puzzle!