What-are-the-last-two-digits-of-2-222-1-

Question Number 133412 by EDWIN88 last updated on 22/Feb/21

What are the last two digits of 2^{222} - 1 ?

Answered by liberty last updated on 22/Feb/21

2^{10} = 1024 \equiv 24 (\mod 100)

2^{20} \equiv 24^{2} = 576 \equiv 76 \equiv - 24 (\mod 100)

Hence 2^{30} \equiv - 24^{2} \equiv - 24 (\mod 100)

2^{40} \equiv - 24 (\mod 100) and so on

Because 222 = 220 + 2; 2^{222} \equiv - 24 \times 2^{2}

\equiv - 96 \equiv 4 (\mod 100)

therefore 2^{222} - 1 \equiv 3 (\mod 100)

The last two digits of 2^{222} - 2 are 03

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