Question

Prove: 24|(5^(2n)-1) for every positive integer n.

Answer 1

And it needs to be proved using induction!

Answer 2

where are u stuck

Answer 3

I showed for n=1 this way:
5^(2*1)-1= 24m
25-1=24m
24=24m
m=1
Since m is an integer, it is true for n =1.
Then I assume it is true for all n = k.
I am stuck on the inductive hypothesis part where i need to prove for n = k+1

Answer 4

assume the statement is true for \(n=k\) :
\(\large 5^{2k} - 1 = 24q\)

you need to provethe statement is true for \(n=k+1\)

Answer 5

for \(n = k+1\) :

\(5^{2(k+1)} - 1 = 5^2(5^{2k} - 1) + 24 = 5^2(24q) + 24 = 24(5^2q+1) = 24r\)

QED.

Answer 6

Yes i proceeded this way:
\[5^{2(k+1)} -1 = 24q\]
\[5^{2k+2}-1=24q\]
\[5^{2k}5^{2}=24q+1\]
\[5^{2k}25=24q+1\]
then i got stuck

Answer 7

oh nevermind! i didn't see your last reply before i sent that! thank you :)

Answer 8

np :) let me knw if smthng is not clear