A cryptogram (part 1)

Logic Level 3

$\begin{array} { l l l l l } & & & A & B \\ \times & & C & D & E \\ \hline F & G & H & I & J \\ \hline \\ \end{array}$

In the above cryptogram, all the letters represent distinct digits from 0 to 9.

Suppose the 5-digit number $\overline{FGHIJ}$ is not a multiple of 9, what is the remainder when $\overline{FGHIJ}$ is divided by 9?

Note that $A, C, F \neq 0$ .

The answer is 4.

1 solution

Chan Lye Lee
May 15, 2017

Let $p = \overline{AB}$ , $q = \overline{CDE}$ and $r = \overline{FGHIJ}$ . Then we have $pq=r$ and $p+q +r = 0+1+2 + \ldots + 9 \equiv 0 \pmod 9$ . Now $pq \equiv -p-q \pmod 9$ , which means that $(p+1)(q+1) \equiv 1 \pmod 9$ .

Using the equations above, we have all the possible values of $p,q$ and $r$ (modulo 9), as follows.

$\begin{array} { l l l l l } & & p & q & r \\ \hline & & 0 & 0 & 0 \\ \hline & & 1 & 4 & 4 \\ \hline & & 2 & - & - \\ \hline & & 3 & 6 & 0 \\ \hline & & 4 & 1 & 4 \\ \hline & & 5 & - & - \\ \hline & & 6 & 3 & 0 \\ \hline & & 7 & 7 & 4 \\ \hline & & 8 & - & - \\ \hline \\ \end{array}$

So $r \equiv 0 \pmod 9$ or $r \equiv \boxed{4} \pmod 9$ .

A cryptogram (part 1)

The answer is 4.

1 solution

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