SOLUTION: can you find by mod (100) the last two number of {{{2^100}}} and {{{2^2006}}}

Question 113235: can you find by mod (100) the last two number of

and

Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
find by mod (100) the last two number of
2^100
= 2^(10^2)
= [2^10]^2
= [1024]^2
= 24^2 (mod 100)
= 576 (mod 100)
= 76 (mod 100)
====================
and
2^2006
2^100 = 76 (mod 100)
2^200 = 5776 = 76 (mod 100)
2^400 = 5776 (mod100) = 76 (mod 100)
2^800 = 2^1600 = 76 (mod 100)
----------
2^2006 = 2^1600*2^400*2^6 = 76*76*64 (mod 100)
= 76*64 mod 100
= 4864 mod 100
= 64 mod 100
================
Cheers,
Stan H.

RELATED QUESTIONS
Given that 79=2 (mod x) where 0 < x <100. Find the set of values for which the statement... (answered by ikleyn)

Determine whether each of the following is true. 1. 7 ≡ 12 mod 5 2. 3 ≡ 24 mod 7... (answered by ikleyn)

Calculate 9^563 mod907 Identify which one of the following options gives the... (answered by ikleyn,math_tutor2020)

MAT 145: Topics In Contemporary Math More Modular Arithmetic Find each of... (answered by ikleyn)

Number 1 A chemist is making a solution that is to be 55% chlorine. He has one... (answered by stanbon)

You can find the percent of growth in Home Depot�s sales from 2002 to 2006, follow these... (answered by jim_thompson5910)

Your last quiz grades were 85, 76, 94 And 100. Set up and solve an equation to find the... (answered by ankor@dixie-net.com)

I would like to know if you can help me with this portion also? Number 1-5 I definitely... (answered by KMST)

when trying to find the percent of an number where do you start. example: (answered by josmiceli)