Product of digits of a 2-digit number is 27. If we add 54 to the number, the new number obtained is a number formed by interchange of the digits. Find the number.

Let's solve this using options verification for rapid identification:
Check Condition 1: Product of digits is 27.

For 39: 3×9=27 (Valid)

For 93: 9×3=27 (Valid)

For 63: 6×3=18
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=27 (Invalid)

For 36: 3×6=18
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=27 (Invalid)

Check Condition 2: Adding 54 reverses the digits.

For 39: 39+54=93.
The digits of 39 reversed are indeed 93. This satisfies the condition perfectly.

For 93: 93+54=147
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=39.

Hence, the number is 39.