mathematics - Find this Unique UVC Palindrome ( ignoring signs and decimal) from Given Fractional Relationship

Given:

U, V, C are three distinct digits ( 0 to 9 ).

UVVVV and CVVVV.U are concatenated numbers.

Dot “.” Stands for decimal.

Relation:

$UVVVV/C= CVVVV.U$

Find U, V , C

Answer

Finding $C$

1. 
   

   
   

   If dividing an integer by $C$ gives a fraction with exactly one digit after the decimal point (note that $U=0$ doesn't work), then $C$ must be non-coprime with $10$, i.e. it must be one of $2,4,5,6,8$.

   
   

   
   


2. 
   

   
   
   

   If $C\geq45$, then the right-hand side is more than $40,000$, and after multiplying by $C$ it won't be a 5-digit number any more. So we must have $C=2$.

   
   

   
   

Finding $U$ and $V$

1. 
   

   
   

   Since $C=2$, the division by $C$ must give $U=5$.

   
   

   
   
   


2. 
   

   
   

   Since $UVVVV$ divided by $2$ is not an integer, $V$ must be odd. Trying the possibilities in turn shows that $V=9$ is the only one which works.

   
   

   
   

Summary

$U=5,V=9,C=2$. The equation is $59999/2=29999.5$.