SOLUTION: Digits a, b, and c can be chosen to make the following multiplication work. What is the 3-digit number <i>abc</i>. <pre> a b c <u>× 2 4</u> 1 c b a 2</pr

 
        a b c
×         2 4
1  c  b  a  2



It's obvious that c must be either 8 or 3, since 8x4=32 and 4x3=12,
the only ways to get a last digit of 2.

We try c=8



Simplify











Divide through by 10



Even if a and b were both the greatest possible digits, which
is 9, the left side would be only 23*9+14*9 = 333, far short
of 1781.  Thus we have ruled out c=8 as possible.  So c=3



Simplify:











Divide through by 10



We solve this Diophantine equation:

Since 14 is the coefficient with the smallest
absolute value, 14, we write the 239 and the
1293 in terms of their closest multiple of 14.

239/14 = 17.071...

So the closest multiple of 14 to 239 is 14*17=238
and so 239 = 238+1

1293/14 = 92.357...

So the closest multiple of 14 to 1293 is 14*92=1288
and so 1293 = 1288+5

So



becomes





Divide every term through by 14





Get all the fractions on the left side,
and other terms on the right side:



The right side equals to an integer, so the left side
must also equal to that same integer.  Let that 
integer be N, so:

 and 





Substitute in 









Since b is a digit, the only integer
N can be is 0. So N=0 and







Substituting N=0 and b=7 in













So a=5, b=7 and c=3

and the three digit number abc is 573

        5 7 3
×         2 4
1  3  7  5  2

Edwin