What-is-the-last-2-digits-of-2-613-

Question Number 14564 by tawa tawa last updated on 02/Jun/17

What is the last 2 digits of 2^{613}

Commented by tawa tawa last updated on 02/Jun/17

God bless you sir .

Commented by tawa tawa last updated on 02/Jun/17

But the question is last two digit

Commented by Tinkutara last updated on 02/Jun/17

2 \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)

2^{613} = 2^{512 + 64 + 32 + 4 + 1}

= (- 4) (16) (- 4) (16) (2) = 8192

\equiv 92 (\mod 100)

Hence last two digits are 92 .

Commented by Tinkutara last updated on 02/Jun/17

2 \equiv 2 (\mod 1000)

2^{2} \equiv 4 (\mod 1000)

2^{4} \equiv 16 (\mod 1000)

2^{8} \equiv 256 (\mod 1000)

2^{16} \equiv - 464 (\mod 1000)

2^{32} \equiv 296 (\mod 1000)

2^{64} \equiv - 384 (\mod 1000)

2^{128} \equiv 456 (\mod 1000)

2^{256} \equiv - 64 (\mod 1000)

2^{512} \equiv 96 (\mod 1000)

2^{613} = 2^{512 + 64 + 32 + 4 + 1}

= (96) (- 384) (296) (16) (2)

\equiv - 808 (\mod 1000)

\equiv 192 (\mod 1000)

Last 3 digits are 192 .

Commented by tawa tawa last updated on 02/Jun/17

i really appreciate sir .

Commented by mrW1 last updated on 02/Jun/17

g o o d w o r k i n g!

p l s f i n d t h e l a s t 3 d i g i t s .

Commented by Tinkutara last updated on 02/Jun/17

I do not know how to find last 3 digits .

Maybe in that case, \mod 1000 is

required . It will be very lengthy and

calculative (use of calculators) .

Commented by RasheedSoomro last updated on 02/Jun/17

e^{x} cellent!

Commented by mrW1 last updated on 02/Jun/17

v e r y g o o d!

p l e a s e t r y t o s o l v e

Q 13724

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