Consider a pair of 3-digit numbers 122 & 213.
Their product is 25986.
Invert both numbers and consider the new product-
it is 68952 i.e. inversion of the previous result!
List all such pairs of 3 digit numbers, avoiding trivial solutions.
Are there such pairs of 4 digit numbers?
Either find some or prove there are none.
(In reply to
Observation by Jer)
No, Jer. Any partial products that have a carry will not have a reversed product when the multiplicands are reversed.
For instance,
211
* 114
--------
844
211x
211xx
=24054
Consider the rightmost carry, in the third digit starting from the right.
8 + 1 + 1 = 10
The product digits to the right of that are 54, the last two digits of the product, which are unaffected by the carry
But the reversed product cannot start with 45, because the carry necessarily increases the the 5 (in this case), changing it to a 6.
No matter the length of the multiplicands, a necessary and sufficient condition for the product of the reversed multiplicands to also be reversed is that there are no carries.
re: Observation
