Digit Divisibility Puzzle: Find the Minimum and Maximum 10-Digit Numbers

A Weekend Challenge from Savvateev

This week’s puzzles come from mathematician Alexey Savvateev — and there are two of them. Before diving in, it’s worth brushing up on one subtlety: division by zero. Remember that zero can be divided by any non-zero number (the result is zero), but dividing by zero is undefined. That detail may turn out to be surprisingly relevant here.

The Digit Divisibility Puzzles

Puzzle 1.
Find the minimum 10-digit integer in which, for every pair of adjacent digits, one digit divides the other.

Puzzle 2.
Find the maximum 10-digit integer consisting of distinct digits in which, for every pair of adjacent digits, one digit divides the other.

Why These Digit Divisibility Problems Are Worth Solving

Both puzzles are classic examples of the extremal principle in olympiad mathematics: instead of finding any solution, you are asked to find the best one — the smallest or the largest. This forces you to think not just about what configurations are valid, but about how to push them to their limits. The divisibility condition on adjacent digits severely restricts which digit sequences are possible, making the search for extremal values a genuine combinatorial challenge.

Try to solve both before looking anything up. The second puzzle, with the additional constraint that all digits must be distinct, is considerably harder.

Source: video lecture