What are the last two digits of 2^(222) −1 ?


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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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