A cryptogram (part 3)

Number Theory Level pending

$\begin{array} { l l l l l } & & 3& 2 & 7 \\ + & & 6 & 5 & 4 \\ \hline & & 9 & 8 & 1 \\ \hline \\ \end{array}$

In the above equation, all the digits from 1 to 9 are used once, where the first 3-digit number is 327, the second 3-digit number is double of the first while the last 3-digit number is three times of the first.

Now consider the following equation: $\begin{array} { l l l l l } & & A& B & C \\ + & & D & E & F \\ \hline & & G & H & I \\ \hline \\ \end{array}$

where different letter represents distinct digit from 1 to 9, $\overline{DEF} = 2\overline{ABC}$ and $\overline{GHI} = 3\overline{ABC}$ .

If $G=6$ , find the value $\overline{GHI}$ .

The answer is 657.

1 solution

Chris Lewis
May 12, 2019

Since $G=6$ , we have $600<\overline{GHI}<699$ . It follows $200<\overline{ABC}<233$ and $400<\overline{DEF}<466$ . So $A=2$ and $D=4$ .

$F$ is even, but isn't any of $\left\{0,2,4,6 \right\}$ so it must be $8$ . $C$ isn't $4$ , so it must be $9$ .

Since $200<\overline{ABC}<233$ , the only possibility is $\overline{ABC}=219$ , which does indeed work, giving $\overline{GHI}=\boxed{657}$ .

A cryptogram (part 3)

The answer is 657.

1 solution

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