Show that the Cryptarithm

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Question:

Show that the Cryptarithm $4 \times \overline{A B}=\overline{C A B}$ does not have any solution.

Solution:

0 is the only unit digit number, which gives the same 0 at the unit digit when multiplied by 4 . So, the possible value of $\mathrm{B}$ is 0 .

Similarly, for A also, 0 is the only possible digit.

But then $A, B$ and $C$ will all be 0 .

And if $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}$ become 0, these numbers cannot be of two $-$ digit or three $-$ digit.

Therefore, both will become a one - digit number.

Thus, there is no solution possible.

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