??7
__x 3??__
?0?3
?1?
__?5?__
?7??3

The above equation was written on a chalkboard to represent an example of a correctly answered multiplication problem. However, the class prankster went when the teacher wasn't looking and smudged many of the numbers. What did the original problem read?

To help understand, let's substitute letters for the question marks.

AB7

x 3CD

-----

E0F3

G1H

I5J

-----

K7LM3

7xD ends in 3, so D=9. AB7x9 must have a 0 in the second place; for B=0, 1, 2, 3, 4, 5, 6, 7, 8, 9 we find A=0, 1, 2, 3, 4, 5, 6, 6, 7, 8 (but the first possibility (A=0) isn't usually allowed).

AB7x3 must be only three digits long, so AB=11 or 22, but only 117x3 has a 5 in the second place.

Finally, 117xC must have a 1 in the second place, so C=1, and we get

117

x 319

-----

1053

117

351

-----

37323