5^5^5.........24times power5 if it is divided by 24 what is the remaindert?

Question

use binomial thoerem

Answer 1

remainder is 1 .

I don't know exactly why :) but with my brain computation , the remainder is 1.

Answer 2

Lets look for a pattern. 5^2 divided by 24 leaves a remainder of 1. 5^3(125) divided by 24 leaves a remainder of 5. 5^4(625) divided by 24 leaves a remainder of 1. Now this looks promising. 5^5 (3125) divided by 24 leaves a remainder of 5. So it appears that if the power of 5 is odd number the remainder is 5 and if the power of 5 is an even number the remainder is 1. Hence the remainder when 5^24 is divided by 24 will be 1.

Answer 3

5^24 = (25)^12 = (25^2)^6 = (24*26 + 1)^6

When this is expanded out all the terms will have 24*26 to some power except the last term--and therefore will be evenly divisible by 24. The last term is 1^6 = 1. The remainder therefore is 1.

Answer 4

Very simple: modulo 24, 5^2 = 25 = 1 (mod 24), so (the next is done modulo 24):

5^24 = (5^2)^12 = 1^12 = 1

the answer is thus 1.

Now, what you wrote is ACTUALLY 5^5^5^5^.... 24 times, and this is VERY different from 5^24, though I doubt you meant this, but nevertheless:
5^5 = [(5^2)^2]*5 = 1^2*5 = 5 (mod 24), and:
(5^5)^5 = 5^5 (mod 24) = 5 (mod 24), so inductively:
5^5^5^....(24 times) = 5 (mod 24)

Regards
Tonio

Answer 5

5^24 / 24
= 24 log 5 - log 24
= 24 (use source) - (use source)
= antilog (use source)