7^{n}-3^{n} is a divisible by 4.

please solve this

it can be prvd byP.M.I.

for n=1

7^{n}-3^{n}=7-3=4 which is div. by 4

so true fr n=1

now,

let it be true for n=k

then,

7^{k}-3^{k}=4m

7^{k}=4m+3^{k}...........(1)

now, for n=k+1

prove it for n=k

pt. 7^{k+1}-3^{k+1} is divisible by 4

now,7^{k+1}-3^{k+1}=7.7^{k}-3^{k+1}

=7(4m+3^{k})-3^{k+1}.........from eq.1

=7.4m+7.3^{k}-3.3^{k}

=7.4m+3^{k}(7-3)

=7.4m+3^{k}.4

=4(7m+3^{k})+0

since remainder =0

so it is divisible by 4

so tru for n=k+1

so

by p.m.i. it is tru for any valueof(n=natural no.)

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