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Prove that 5^{k}-5 is divisible by 4 by principle of mathematical induction

^{n}-5

For n=1,

p(1): 5-5 = 0 which is divisible by 4.

therefore p(1) is divisible by 4.

now, suppose p(k) is divisible by 4; k belongs to N.

P(k): 5

^{k}-5 = 4*m ; m belongs to N. _________ [1]

now, we have to prove p(k+1) is divisible by 4.

P(k+1): 5

^{k+1}-5

= 5

^{k}*5 - 5

= (4*m+5)5 - 5 <from [1]>

= 5{(4*m+5)-1}

= 5{4*m+5-1}

= 5{4*m+4}

= 5{4(m+1)}

= 4(5*m+5) which is divisible by 4.

p(k) is divisible by 4 which implies p(k+1) is divisible by 4.

hence, p(n) is divisible by 4, by PMI.