whats the remainder ?

Question

whats the remainder when 13^400 is divided by 187 ?

Answer 1

13^20 (mod 187) = 1, so
(13^20)^20 (mod 187) = 1.

Answer 2

Given: 13^400 ÷ 187 = X∙R

=(ln 13^400) / ln 187
= 400 ln 13 / ln 187
= (1025∙979743..) / (5.231108617..)
= 196∙1304607
Taken the decimal as the remainder and getting the Anti log:
= e^0∙1304607
= 1∙139353162 Remainder.

Answer 3

You need to do the arithmetic (mod 187). Modular arithmetic is in fact the arithmetic of remainders. The relevant rule (which you can probably prove for yourself) is that the remainder of the product of two numbers (or power of one number) is equal to the product of the remainders (or power of the remainder).

Example: Find the remainder of 17^6 divided by 12. The remainder of 17/12 is 5, 5*5=25=-1 mod 12, 5*5*5*5=-1*-1=+1 mod 12, 5^6=-1*-1*-1=-1 mod 12. So the answer is -1.

Answer 4

187=11X17

by fermats rule

13^10=1mod11
hence 13^400=1^40mod 11
or 1 mod 11

13^16=1mod17
hence 13^400=1^25mod 17
or 1 mod 17

hence 13^400 = 1 mod 187

Answer 5

Actually 13^400 means 13 to the 400th power. The only program that I know that should actually give that number in all it's digits, and I am not sure of this, because i am using my son-ii-law's Windows machine, is the progam dc in Linux. if you need a 1,000 digits it gives them.

A few of the live CD distros have dc installed, so one could boot on a live CD and calculate stuff like this, without installing anything on your HD.

I use dc for my checkbooks, because it leaves all the digits used on the screen so I can verify the numbers.

Answer 6

There's nothing such as 13^400... :S