Many people sent correct answers, which I had expected. However, very few sent the explanation behind the answers, which is what I was looking for. Some of you used equations to solve the question. Intent was to solve the questions with no equations, and only use logical reasoning.
Ones to send the correct answer included Ajay Pandit (Bangalore), Anshi Aggarwal (Gurgaon), Radhika Goyal (New York), Karan Sharma (Jaipur), Pravin Panchagnula and Rhea Mittal (Mumbai).
Here is a way to think about both the questions without equations:
(i) The number is greater than 9 since there are two digits
(ii) It is less than 23 because 23 x 4 1/2 is greater than 100
(iii) It is an even number because it is an integer when multiplied by 4 1/2
(iv) Since the number is even, its half is an integer. When multiplied by 9, it yields the reverse and hence the reverse is divisible by 9
(v) Since the number has the same digits as the reverse, it implies that the number itself is divisible by 9
(vi) between 9 and 23, only number divisible by 9 is 18 which is the answer
(i) None of the four integers is a 10 or a multiple of 5, if it was the product would end in a 0
(ii) At least one of the digits is less than 10, or the product would be 5 digits
(iii) All four are less than 10 [because of reason #(i)]
(iv) Therefore the only possibilities are 1,2,3,4 or 6,7,8,9. Since 1,2,3,4 yields only 24, 6,7,8,9 is the answer!
Hope you enjoyed the puzzle!