Divisibility Sequences

Masum Billal () and Samin Riasat

Chapter Chapter 3 in Integer Sequences, 2021, pp 57-71 from Springer

Abstract: Abstract Consider the classic problem that the product of n consecutive integers is divisible by n!. The proof of this fact is the basis of our study on this topic. A beginner usually tries to prove this with some basic modular arithmetic, for example, at least one of the n consecutive integers is divisible by n since each of them leaves a different remainder upon division by n. Similarly, at least one of those integers is divisible by i for $$1\le i\le n$$ 1 ≤ i ≤ n . However, this does not prove that the product of all $$1\le i\le n$$ 1 ≤ i ≤ n divides n! as well, although the least common multiple of them $${{\,\mathrm{lcm}\,}}(1,2,\ldots ,n)$$ lcm ( 1 , 2 , … , n ) does.

Date: 2021
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DOI: 10.1007/978-981-16-0570-3_3

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