2022 CSAT — Q10

We need a $3$-digit $N$ such that $N$, $2N$, $3N$ together use each of the digits $1$–$9$ exactly once (pandigital, no zeros). Since $3N$ must stay $3$-digit, $N \le 333$.

The complete list of such $N$ is the classic set:

Each verified pandigital. Now apply the extra rule: the first row $N$ must use only digits from $\{2,3,7,9\}$ (exactly three of those four):

• $192$: digits $\{1,9,2\}$ — contains $1$. Rejected.
• $219$: digits $\{2,1,9\}$ — contains $1$. Rejected.
• $273$: digits $\{2,7,3\}$ — all in $\{2,3,7,9\}$. Valid.
• $327$: digits $\{3,2,7\}$ — all in $\{2,3,7,9\}$. Valid.

Exactly two first-row numbers ($273$ and $327$) satisfy the constraint.

Answer: (c) 2.

The hidden work is knowing — or quickly reconstructing — the four pandigital $\{N,2N,3N\}$ triples. Build them by the constraints $N\le 333$, no repeated/zero digits across all nine cells; the search is short because $2N,3N$ being $3$-digit pins $N$ between $100$ and $333$. The restriction “first row from $\{2,3,7,9\}$” is then a pure filter that eliminates the two numbers carrying a $1$.