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Question Number 205032 by BaliramKumar last updated on 06/Mar/24

Commented by BaliramKumar last updated on 06/Mar/24

please clarification
	Answered by Rasheed.Sindhi last updated on 06/Mar/24

	$N = q D + 14 w h e r e D > 14$ $3 N = 3 q D + 42 = p D + 8$ $4 N = 4 q D + 56 = p D + 8 + q D + 14$ $4 q D + 56 = (p + q) D + 22$ $(p + q) D - 4 q D = 56 - 22 = 34$ $D (p - 3 q) = 34$ $D_{> 14} = \frac{34}{p - 3 q}$ $p - 3 q = 1, 2, 17, 34$ $D = 34, 17, \overset{\times}{2}, \overset{\times}{1}$ $∵ D > 14 ∴ D = 34, 17$ $C a s e 1 : D = 17$ $N = 17 q + 14 \equiv 14 (m o d 17)$ $3 N = 51 q + 42 \equiv 8 (m o d 17)$ $4 N = 4 \times 17 q + 56 \equiv 5$ $C a s e 2 : D = 34$ $N = 34 q + 14 \equiv 14 (m o d 34)$ $3 N = 3 \times 34 q + 42 \equiv 8 (m o d 34)$ $4 N = 4 \times 34 q + 56 \equiv 22 (m o d 34)$
	Answered by A5T last updated on 06/Mar/24

	$N = D k + 14; 3 N = D q + 8; 4 N = D c + r$ $D ⩾ 15$ $3 N = 3 D k + 14 \times 3 = 3 D k + 34 + 8$ $\Rightarrow D ∣ 34 a n d D ⩾ 15 \Rightarrow D = 34 o r 17$ $4 N = D (k + q) + 22; I f D = 34; t h e n r = 22$ $B u t i f D = 17, t h e n 4 N = D (k + q + 1) + 5 \Rightarrow r = 5$ $\Rightarrow r = 5 o r r = 22$
	Commented by BaliramKumar last updated on 06/Mar/24

	$Q say proper divisor$
	Commented by A5T last updated on 06/Mar/24

	$T h e n u m b e r, D, c a n n o t b e a p r o p e r ‘ ‘ {d i v i s o r}^{″},$ $o t h e r w i s e {i t}^{'} d d i v i d e N c o m p l e t e l y w i t h o u t a$ $r e m a i n d e r .$
	Commented by BaliramKumar last updated on 06/Mar/24

	$thanks$
	Answered by Rasheed.Sindhi last updated on 06/Mar/24

	$N \equiv 14 (m o d D) \Rightarrow D > 14$ $3 N \equiv 42 \equiv 8 (m o d D)$ $D ∣ (42 - 8) \Rightarrow D ∣ 34 \land D > 14$ $D = 17, 34$ $N = 17 q_{1} + 14, 34 q_{1} + 14$ $C a s e 1 : D = 17$ $N = 17 q_{1} + 14 \equiv 14 (m o d 17)$ $4 N = 4 \times 17 q_{1} + 56 \equiv 56 (m o d 17)$ $4 N \equiv 56 - 3 \times 17 = 5 (m o d 17$ $C a s e 1 : D = 34$ $N = 34 q_{2} + 14 \equiv 14 (m o d 34)$ $4 N = 4 \times 34 q_{1} + 56 \equiv 56 (m o d 34)$ $4 N \equiv 56 - 34 = 22 (m o d 34)$
	Commented by BaliramKumar last updated on 06/Mar/24

	$thanks$
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