Q:

It's my sister's birthday

I'd ordered her a cake, but when I got to the bakery to pick it up, the cake decorator had transposed the digits of her age.
"She'll thank you for the compliment," I said, "but her age is a prime number, and the number on the cake is the product of the digits of her age at not one, but two future birthdays."
"You don't think she'll break any longevity records, then?" the cake decorator shouted after me as I was leaving the shop without my cake.
How old is my sister?

A:

Solution
Your sister is

 53,

and the age on the cake is

 35, equal to the digit product of both 57 and 75 (assuming she won't reach 157).

Proof of uniqueness
We're given several pieces of information:

"transposed the digits of her age" - her age is 2 digits.
"She'll thank you for the compliment" - the age on the cake makes her seem younger, so the second digit is smaller than the first.
"her age is a prime number" - her age is a 2-digit prime (only 20 of those!).

Taking these pieces of information into account, we've already narrowed it down to a small set of possibilities:

 41, 43, 53, 61, 83.

The reversed numbers are:

 14=2*7, 34=2*17, 35=5*7, 16=4*4=2*8, 38=2*19.

We can immediately rule out

 43 and 83 because 17 and 19 are not digits, and 41 and 61 because only 72 and 82 (respectively) are larger with the right digit product.

That leaves only one possible answer.

A:

She was

 53

The decorator put

 35  which is the product of digits for 57 and 75.  If she set a longevity record and reached 157, it would be that, too.

