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

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.

Solution

Dudeney offers up $3,333,377,733$.

This is divisible by $3$ and $7$, and the sum of its digits is $42$, also divisible by $3$ and $7$.

In his words:

There must be at fewest three $7$'s and seven $3$'s, and the $7$'s must be placed as far to the right as possible.

However, we also offer up $777$ which equally well fulfils the conditions of the question.

Sources

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