Problems & Puzzles: Puzzles

 Puzzle 396. X*R=Y Digging around the always interesting pages from Mr. Leonid Mochalov, I came to face the following puzzle "Two cofactors". Here Leonid asks to find an integer solution to the product of two factors one of which is integer while the other is rational: X*R=Y, where: - X & Y are integers - R is rational, non integer - all the digits 1 to 9 must be used once in the pair X & R. Leonid gives the following solution, without any comment: 64 * 3.921875 = 251 Q1. Can you explain why (perhaps) Leonid gave this solution? By my own I produced other solutions around the Leonid's model from which I show you just three of them, that perhaps you will appreciate as interesting, too: 192 * 3 . 46875 = 666 1306728 * 94 . 5 = 123485796 873024 * 961 . 5 = 839412576 Q2. Can you produce other interesting solutions around the Leonid's model? (feel you free to make the slight/interesting variations you like to the given model by Leonid)

Contribution came from J C Rosa

About the questions 1  & 2:

In the Leonid's model we have X=2^n , R is a decimal number and Y is a prime number .

Here is a method ( perhaps the Leonis method ?) for obtain all solutions of this model :
1) Y=5
2) X=2
3) R=Y/X
4) if R<1 then take for Y the next prime and go back to 2)
5) if in the pair X,R the nine digits from 1 to 9 are used then the triplet X,R,Y is a solution
6) multiply X by 2 and go back to 3)

With this method I obtained 1360 solutions .
The earliest is the Leonid'solution and the greatest that I found is :
2*9876134,5=19752269

Among these 1360 solutions I have found only one solution where Y is a palprime . Here is it:

4*896713,25=3586853
...

With my model where R=P/Q ( Q is not a divisor of P but obviously a divisor of X )

I found several palindromic numbers for Y . Among these solutions here are two curios
and nice  ( I  think...) ( I have only cherched with Q=7).

63*(1480259/7)=13322331            63*(2590148/7)=23311332
63*(148259/7)=1334331                63*(259148/7)=2332332

X=14 ; R=235689/7  ; Y=471378  (  14*(235689/7)=471378 )
X=26 ; R=45789/13  ; Y=91578  (  26*(45789/13)=91578 )

...

What about the following model:  R1*R2=Y ?

No problem . There are many solutions and I have found the following solutions where Y
is a palindromic number :
(56/3)*(18294/7)=48784
(56/3)*(94182/7)=251152

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