## Ken's POTW

Reversed Products
1. Solve for A,B,C,D,E,F as unique digits 0-9: ABCDE x F = EDCBA
2. In Y x k = Z, k is a single digit from 2-9, and Z has the same digits as Y in reverse order (i.e. if Y = 241456, then Z = 654142.) Find all combinations (Y,k) that solve the equation.
It may seem like the second problem is much too undefined. Actually, the solutions can be reduced to some simple descriptions.

Source: Original, though I wouldn't be surprised to find these printed somewhere else.

Solutions were received from: Radu Ionescu, Walter Prager, and Philippe Fondanaiche (Paris). Phillippe's solution succinctly shows the complete answer:
```>From Philippe Fondanaiche Paris France

Good evening,

Considering the general equation Y x k = Z, we have the following properties:

1) Whatever Y, k can take only the values 4 and 9. Indeed, taking into
consideration the first digit A and the last one N in Y, we have :
A = last digit of Nxk and N = Axk + r (carry over < k)
So A = last digit of Axk^2 + rxk  (a)
It is easy to check that for k = 2,3,5,6,7,8 there is no couple (A,r)
satisfying the identity (a).
For k = 4, there is only one couple A = 2 and N = 8 and for k = 9 there is
also a single couple A = 1 and N = 9.

2) For k = 4, the lowest value of Y is 2 178 and for k = 9, the lowest one is
1 089
with the relations 2 178 x 4 = 8 712 and 1 089 x 9 = 9 801.
In the set of the 4 digit numbers there is no other solution (easy to
demonstrate).

3) Starting from these lowest values of Y, we build new solutions with p
digit numbers Y by introducing p-4 times the number 9 between 21 and 78 or
between 10 and 89.
For example 21 978 x 4 = 87 912 :this the answer of the question 1 with 5
different digits for Y.
Other example : 1 099 989 x 9 = 9 899 901.
So there is an infinite number of solutions such as 219.....978 or
109......989.

4) By juxtaposing 2 or more identical solutions above mentioned, we create a
set of new solutions which can be extended up to the infinite.
For example 2 197 821 978 x 4 = 8 791 287 912 or 10 891 089 x 9 = 98 019 801

Best regards.
```

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