Question

Prove that $5^k-5$ is multiple of 4

Answer 1

Let P(n): $5^N-5$
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): $5^k-5=4^{x}m$; m belongs to N. - - - -- - (1)
Now, we have to prove p(k+1) is divisible by 4.
P(k+1): $5^{k+1}-5$
$=5^{k+5}-5$
$=(4^{x}m+5)5-5$ <from [1]>
= 5{($4^x$+5)-1}
= 5{$4^x$m+5-1}
= 5{$4^x$ m+4}
= 5{4(m+1)}
=4 ($5^x$ 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