Henry Ernest Dudeney/Modern Puzzles/68 - Threes and Sevens

Modern Puzzles by Henry Ernest Dudeney: $68$

Threes and Sevens

What is the smallest number composed only of the digits $3$ and $7$ that may be divided by $3$ and $7$,

and also the sum of its digits by $3$ and $7$, without any remainder.

Thus, for example, $7 \ 7 \ 3 \ 3 \ 7 \ 3 \ 3$ is divisible by $3$ and $7$ without remainder,

but the sum of its digits ($33$), while divisible by $3$, is not divisible by $7$ without remainder.

Therefore it is not a solution.

Sources

• 1926: Henry Ernest Dudeney: Modern Puzzles ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $68$. -- Threes and Sevens
• 1968: Henry Ernest Dudeney: 536 Puzzles & Curious Problems ... (previous) ... (next): Arithmetical and Algebraical Problems: Digital Puzzles: $119$. Threes and Sevens