How-to-calculate-the-last-two-digits-of-2-576-

Question Number 15770 by arnabpapu550@gmail.com last updated on 13/Jun/17

How to calculate the last two digits of 2^{576}

Answered by Tinkutara last updated on 14/Jun/17

2^{576} = 2^{4 \times 144} \equiv 6

Last digit is 6 .

2^{1} \equiv 2 (\mod 100)

2^{2} \equiv 4 (\mod 100)

2^{4} \equiv 16 (\mod 100)

2^{8} \equiv - 44 (\mod 100)

2^{16} \equiv 36 (\mod 100)

2^{32} \equiv - 4 (\mod 100)

2^{64} \equiv 16 (\mod 100)

2^{128} \equiv - 44 (\mod 100)

2^{256} \equiv 36 (\mod 100)

2^{512} \equiv - 4 (\mod 100)

Now 2^{576} = 2^{512 + 64} \equiv (- 4) (16) (\mod 100)

\equiv - 64 (\mod 100) \equiv 36 (\mod 100)

Last two digits are 36 .

Commented by arnabpapu550@gmail.com last updated on 14/Jun/17

What is (\mod 100) ?

Commented by arnabpapu550@gmail.com last updated on 14/Jun/17

How to calculate 3^{576} ?

Commented by Tinkutara last updated on 14/Jun/17

Remainder when a number is divided

by 100 . You can proceed in the same

way for 3^{576} .

Commented by arnabpapu550@gmail.com last updated on 15/Jun/17

Now I am trying to calculate last two digits

of 3^{576}

3^{1} \equiv 3 (\mod 100)

3^{2} \equiv 9 (\mod 100)

3^{4} \equiv - 19 (\mod 100)

3^{8} \equiv - 39 (\mod 100)

3^{16} \equiv 21 (\mod 100)

3^{32} \equiv 41 (\mod 100)

3^{64} \equiv - 19 (\mod 100)

3^{128} \equiv - 39 (\mod 100)

3^{256} \equiv 21 (\mod 100)

3^{512} = 41 (\mod 100)

∴ 3^{576} = 3^{512 + 64} = 41 \times (- 19) (\mod 100)

= 21

Is it right ?

Commented by arnabpapu550@gmail.com last updated on 14/Jun/17

How can you get 2^{256}, 2^{512} etc ?

Commented by Tinkutara last updated on 15/Jun/17

Square the remainders obtained in

previous step and find their remainders

when divided by 100 . For easy

calculation and simplification, if the

remainders are greater than 50,

subtract 100 from them and put a minus

sign . Now repeat the same procedure

again and again .

Commented by Tinkutara last updated on 15/Jun/17

Yes!

Commented by arnabpapu550@gmail.com last updated on 15/Jun/17

Thank you very much .