Find the remainder when:− (a) 41! is divided by 1681 (b) 225! is divided by 227 (c) 15! is divided by 19


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Question Number 167110 by nimnim last updated on 06/Mar/22

$F i n d t h e r e m a i n d e r w h e n : -$ $(a) 41! i s d i v i d e d b y 1681$ $(b) 225! i s d i v i d e d b y 227$ $(c) 15! i s d i v i d e d b y 19$
	Answered by Rasheed.Sindhi last updated on 06/Mar/22

	$\begin{matrix} p \in P \Leftrightarrow (p - 1)! \equiv - 1 [p] \end{matrix}$ $(b)$ $(227 - 1)! \equiv - 1 [227] [∵ 227 \in P]$ $226! \equiv - 1 [227]$ $226! \equiv - 1 + 227 [227]$ $226! \equiv 226 [227]$ $225! \equiv 1 [227]$ $(c)$ $(19 - 1)! \equiv - 1 [19] [∵ 19 \in P]$ $18! \equiv - 1 + 19 [19]$ $18! \equiv 18 [19]$ $17! \equiv 1 [19]$ $17! \equiv 1 + 19 \times 8 [19]$ $17! \equiv 153 [19]$ $16! \equiv 9 [19]$ $16! \equiv 9 + 19 \times 13 [19]$ $16! \equiv 256 [19]$ $15! \equiv 16 [19]$
	Commented by nimnim last updated on 06/Mar/22

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