5

Part of 2024 Polish Junior MO Finals

Problems(1)

A huge number with digits 1 and 9 is a multiple of 19

Source: 2024 Polish Junior Math Olympiad Finals P5

S=\underbrace{111\dots 1}_{19}\underbrace{999\dots 9}_{19}

. Show that the

2S

\underbrace{111\dots 1}_{S}\underbrace{999\dots 9}_{S}

is a multiple of

19

number theorynumber theory proposedDivisibilitymultipleDigits