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Consecutive string with a nice property

Source: IMSC 2024 Day 1 Problem 1

June 28, 2024
number theory unsolvednumber theorynumber theory proposedDivisibility

Problem Statement

For a positive integer nn denote by P0(n)P_0(n) the product of all non-zero digits of nn. Let N0N_0 be the set of all positive integers nn such that P0(n)∣nP_0(n)|n. Find the largest possible value of ℓ\ell such that N0N_0 contains infinitely many strings of ℓ\ell consecutive integers.
Proposed by Navid Safaei, Iran
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