There are infinitely many pairs of numbers whose sum equals their product.

However, there is only one solution to the equation below, in which each letter stands for a single, distinct digit.

AB × C.DE = AB + C.DE

What digit does each letter in this equation represent?

For the sum & product of two numbers to be the same, if the first is x the second is of the form x/(x-1) or 1+1/(x-1). This is easily proved.
In this case, x is a 2 digit number (AB) & C=1
1/(x-1) = y/100 (where y is a one or two digit number). This means that (x-1) divides 100. Since (x-1) cannot have more than two digits it does not equal 100. Check out the other possibilities
x has 2 digits so (x-1) cannot be 2,4 or 5
(x-1)=50 implies that x=51 (oops: B=1=C)
(x-1)=20 - same problem: B=1=C
This leaves x-1 = 25; so x=26 & 1/(x-1) = 0.04
So: A=2; B=6; C=1; D=0; E=4

Posted by DrBob on 2003-10-31 16:16:51