If a is a single digit number and a”-a is divisible by
10, where n is any natural number, how many values
can a take?<

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1. Answer:

   If a is a single digit number and a^n – a is divisible by 10, where n is any natural number, how many values can a take?

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   Vijay Mankar (विजय मानकर)

   Updated Wed

   For 10|an−a

   10∣a5−a,∀a={1,2,3,4,5,6,7,8,9}

   Since, 15−1=0,25−2=30,35−3=240,45−4=1020,55−5=3120,65−6=7770,75−7=16800,95−9=59040

   Reason:

   ϕ(10)=10(1−1/2)(1−1/5)=4

   By Euler theorem,

   a4≡1(mod10)

   a4−1≡0(mod10)

   a(a4−1)=a5−a≡0(mod10)

   So possible values of a are a={1,2,3,4,5,6,7,8,9} for different n.

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