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Question Number 205175 by BaliramKumar last updated on 12/Mar/24

	Answered by Rasheed.Sindhi last updated on 12/Mar/24

	$413283 P 759387$ $∙ D i v i d e t h e n u m b e r i n g r o u p s o f 3$ $f r o m r i g h t s i d e$ $4, 132, 83 P, 759, 387$ $∙ A p p l y s u b t r a c t i o n a n d a d d i t i o n$ $a l t e r n a t i v e l y b e t w e e n t h e g r o u p s :$ $387 - 759 + 83 P - 132 + 4$ $T h e n u m b e r i s d i v i s i b l e b y 13 i f$ $t h e r e s u l t i s 0 o r a m u l t i p l e o f 13$ $∵ 13 ∣ (387 - 759 + 83 P - 132 + 4)$ $o r 13 ∣ (387 - 759 + (830 + P) - 132 + 4)$ $o r 13 ∣ 330 + P$ $o r 13 ∣ 13 \times 25 + 5 + P$ $o r 13 ∣ 5 + P$ $R e c a l l t h a t 0 ⩽ P ⩽ 9 a l s o .$ $∴ P = 8$
	Commented by BaliramKumar last updated on 12/Mar/24

	$Thanks sir$
	Answered by A5T last updated on 12/Mar/24

	$[3^{3 k + 1} \overset{13}{\equiv} 3; 3^{3 k + 2} \overset{13}{\equiv}; 3^{3 k} \overset{13}{\equiv} 1]$ $4 (10^{12}) + 1 (10^{11}) + 3 (10^{10}) + . . . + 3 (10^{2}) + 8 (10) + 7$ $\overset{13}{\equiv} 4 {(- 3)}^{12} + {(- 3)}^{11} + 3 {(- 3)}^{10} + . . . + 3 {(- 3)}^{2} + 8 (- 3) + 7$ $\equiv 4 (1) + 1 (- 9) + 3 (3) + 2 (- 1) + 8 (9) + 3 (- 3) + P$ $+ 7 (- 9) + 5 (3) + 9 (- 1) + 3 (9) + 8 (- 3) + 7 = P + 18$ $\overset{13}{\equiv} 0 \Rightarrow P \equiv - 18 \equiv - 5 \equiv 8 (m o d 13) \Rightarrow P = 8$
	Commented by BaliramKumar last updated on 12/Mar/24

	$thanks sir$ $what is osculator how it works$
	Commented by A5T last updated on 12/Mar/24

	$I f y o u w a n t t o f i n d t h e o s c u l a t o r o f a n u m b e r$ $t h a t i s c o - p r i m e w i t h 10, s a y 17 o r 19, f i n d t h e$ $s m a l l e s t m u l t i p l e o f t h e n u m b e r, s a y 17 : (51)$ $o r 19 : (19), t h a t i s \underline{+} 1 m o d 10 .$ $I f t h e s m a l l e s t m u l t i p l e t h a t i s \underline{+} 1 m o d 10 i s$ $a c t u a l l y + 1 (m o d 10) a s i n t h e c a s e o f 17, t h e f i r s t$ $d i g i t o f t h a t m u l t i p l e i s c a l l e d t h e ‘ ‘ {n e g a t i v e}^{″}$ $o s c u l a t o r o f t h e n u m b e r .$ $S o, i f y o u w a n t t o c h e c k t h e d i v i s i b i l i t y o f a n u m b e r$ $b y 17, s a y 3281, t h e n y o u ‘ ‘ s u b t r a c t 5 {t i m e s}^{″} t h e$ $l a s t d i g i t, 1, f r o m t h e r e m a i n i n g p a r t : 328 - 5 \times 1$ $= 323 \overset{\underset{o p e r a t i o n}{s a m e}}{\to} 32 - 3 \times 5 = 17 . S i n c e 17 i s d i v i s i b l e b y 17,$ $3281 m u s t a l s o b e d i v i s i b l e b y 17 .$ $F o r a n u m b e r l i k e 19 \equiv - 1 (m o d 10); t h e ‘ ‘ {p o s i t i v e}^{″}$ $o s c u l a t o r i s t h e f i r s t d i g i t o f 19 + 1, w h i c h i s 2,$ $s o i n s t e a d o f ‘ ‘ {s u b t r a c t i n g}^{″} t h e l a s t d i g i t, y o u$ $a d d i t . F o r e x a m p l e, 3876 : 387 + 6 \times 2 = 399$ $\to 39 + 9 \times 2 = 57 = 19 \times 3 (d i v i s i b l e b y 19) \Rightarrow 19 ∣ 3876$
	Commented by A5T last updated on 12/Mar/24

	$T h e n e g a t i v e o s c u l a t o r o f 7 i s 2$ $F i r s t m u l t i p l e s o f 7 : 7, 14, 21 : + 1 (m o d 10)$ $S o, o s c u l a t o r i s f i r s t d i g i t o f 21 - 1$ $T h e p o s i t i v e o s c u l a t o r o f 13 i s 4$ $F i r s t m u l t i p l e s o f 13 : 13, 26, 39 : - 1 (m o d 10)$ $S o, o s c u l a t o r i s f i r s t d i g i t o f 39 + 1$
	Commented by BaliramKumar last updated on 12/Mar/24

	$thanks sir$
	Answered by A5T last updated on 12/Mar/24

	$[10^{3} \overset{13}{\equiv} - 1; 10^{6} \overset{13}{\equiv} 1]$ $413283 P 759387 = 387 + 759 \times 1000 + 83 P \times 10^{6}$ $+ 132 \times 10^{9} + 4 \times 10^{12} \equiv 387 - 759 + 83 P - 132 + 4$ $\overset{13}{\equiv} 0 \Rightarrow 83 P - 500 \overset{13}{\equiv} 0 \Rightarrow 830 + P \equiv 500 \equiv 6 (m o d 13)$ $11 + P \overset{13}{\equiv} 6 \Rightarrow P \equiv - 5 \overset{13}{\equiv} 8$
	Answered by Rasheed.Sindhi last updated on 13/Mar/24

	$413283 P 759387$ $= 4132830000000 + P 759387$ $O s c u l a t o r o f 13 = 4$ $P 759387$ $\begin{array}{ccc} P & 7 & 5 & 9 & 3 & 8 & 7 \\ 4 \times 4 + 2 + P & 4 \times 4 + 1 + 7 & 4 \times 2 + 1 + 5 & 4 \times 0 + 3 + 9 & 4 \times 6 + 3 + 3 & 4 \times 7 + 8 \\ 18 + P & 24 & 14 & 12 & 30 & 36 \end{array}$ $4132830000000$ $\begin{array}{ccc} 4 & 1 & 3 & 2 & 8 & 3 & 0 & . . . & 0 \\ 4 \times 8 + 3 + 4 & 4 \times 9 + 1 + 1 & 4 \times 4 + 3 & 4 \times 0 + 2 + 2 & 4 \times 3 + 8 & 4 \times 0 + 3 & 4 \times 0 + 0 \\ 39 & 38 & 19 & 4 & 20 & 3 & 0 \end{array}$ $13 ∣ (18 + P + 39)$ $13 ∣ (5 + P)$ $P = 8$
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