How many numbers x are found between 9000 and 11000 and verify x=12[19], x=5[13], x=9[11] ?


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Question Number 152139 by Ar Brandon last updated on 26/Aug/21

$How many numbers x are found between 9000 and 11000$ $and verify x = 12 [19], x = 5 [13], x = 9 [11] ?$
Commented by Rasheed.Sindhi last updated on 26/Aug/21

$One (x = 10899)$
	Answered by Rasheed.Sindhi last updated on 26/Aug/21

	$x \equiv 12 [19], x \equiv 5 [13], x \equiv 9 [11]$ $x \equiv 12 + 19 [19], x \equiv 5 + 13 [13], x \equiv 9 + 11 [11]$ $x \equiv 31 [19], x \equiv 18 [13], x \equiv 20 [11]$ $x \equiv 31 + 19 [19], x \equiv 18 + 13 [13], x \equiv 20 + 11 [11]$ $x \equiv 50 [19], x \equiv 3113], x \equiv 31 [11]$ $S m a l l e s t v a l u e o f x : 31 w h i c h s a t i s f y$ $a l l t h r e e c o n g r u e n c e .$ $n e x t s u c h v a l u e 31 + LCM (11, 13, 19)$ $= 31 + 2717 = 2748$ $I n g e n e r a l 31 + n LCM (11, 13, 19)$ $T o f i n d least v a l u e o f n$ $n = ⌊ \frac{9000}{2717} ⌋ = 3 or n = ⌊ \frac{9000}{2717} ⌋ + 1 = 4$ $31 + 3 \times 2717 = 8182 < 9000 (\times)$ $31 + 4 \times 2717 = 10899 (✓)$ $31 + 5 \times 2727 = 13616 > 11000 (\times)$ $ONE ONLY$
	Commented by Ar Brandon last updated on 26/Aug/21

	$OK Sir Thanks$
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