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# Show that cryptarithm $4 \times \overline{AB}=\overline{CAB}$ does not have any solution.

To do:

We have to show that cryptarithm $4 \times \overline{AB}=\overline{CAB}$ does not have any solution.

Solution:

$4 \times \overline{\mathrm{AB}}=\overline{\mathrm{CAB}}$

$\overline{\mathrm{AB}}$ is a two digit number and $\overline{\mathrm{CAB}}$ is a three digit number.

$4 \times \overline{\mathrm{AB}}=\overline{\mathrm{CAB}}$

This implies,

$ A\ B$

$\times 4$

--------------

$C\ AB$

This implies,

$4 \times B$ is a number whose units digit is $B$.

There is no such digit which when multiplied by 4 has the same number at the units place.

Hence, the given cryptarithm has no solution.

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