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Six digit number divisible by 271

Source: Bundeswettbewerb Mathematik 2019, Round 1 - Problem 2

August 5, 2019

DigitsdivisibleDivisibilitynumber theory

Problem Statement

The lettes

A,C,F,H,L

and

S

represent six not necessarily distinct decimal digits so that

S \ne 0

and

F \ne 0

. We form the two six-digit numbers

SCHLAF

and

FLACHS

.
Show that the difference of these two numbers is divisible by

271

if and only if

C=L

and

H=A

.
Remark: The words "Schlaf" and "Flachs" are German for "sleep" and "flax".