how many solution the equation ⌊ x ⌋ +2016. {x} = 38?


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Question Number 90698 by jagoll last updated on 25/Apr/20

$h o w m a n y s o l u t i o n t h e e q u a t i o n$ $⌊ x ⌋ + 2016 . {x} = 38 ?$
Commented by mr W last updated on 25/Apr/20

$i n g e n e r a l$ $⌊ x ⌋ + n {x} = m$ $h a s n s o l u t i o n s .$
Commented by jagoll last updated on 25/Apr/20

$i f c h a n g e n ⌊ x ⌋ + {x} = m$ $h o w s i r ?$
Commented by mr W last updated on 25/Apr/20

$i t h a s n o t m u c h s e n s e .$ ${x} m u s t b e z e r o . i . e . x m u s t b e i n t e g e r .$ $⌊ x ⌋ = x$ $n x = m$ $i f m \equiv 0 m o d (n) t h e n o n e s o l u t i o n :$ $x = \frac{m}{n}$ $o t h e r w i s e n o s o l u t i o n .$
	Answered by mr W last updated on 25/Apr/20

	$x = ⌊ x ⌋ + {x} = n + f$ $n + 2016 f = 38$ $0 ⩽ f = \frac{38 - n}{2016} < 1$ $\Rightarrow 0 ⩽ 38 - n < 2016$ $\Rightarrow - 1978 < n ⩽ 38$ $\Rightarrow - 1977 ⩽ n ⩽ 38$ $i . e . t h e r e a r e 2016 s o l u t i o n s .$
	Commented by jagoll last updated on 25/Apr/20

	$w h y s i r h a s 2016 s o l u t i o n ?$
	Commented by mr W last updated on 25/Apr/20

	$f o r e a c h n \in [- 1977, 38] t h e r e i s a$ $s o l u t i o n x = n + \frac{38 - n}{2016} .$ $w e h a v e t o t a l l y 2016 v a l u e s o f n :$ $- 1977, - 1976, . . ., - 1, 0, 1, . . ., 38$ $t h e r e f o r e t h e r e a r e 2016 s o l u t i o n s$ $f o r ⌊ x ⌋ + 2016 {x} = 38 .$
	Commented by jagoll last updated on 25/Apr/20

	$o o i t d o e s m e a n$ $- 1977, - 1976, . . ., 38 e q u a l t o$ $A P w i t h d = 1 a n d T_{1} = - 1977$ $s o 38 = - 1977 + (n - 1) \times 1$ $38 = n - 1978$ $n = 38 + 1978 = 2016 s i r ?$
	Commented by mr W last updated on 25/Apr/20

	$y e s .$ $f r o m - 1977 t o 38 t h e r e a r e 2016$ $i n t e g e r s .$
	Commented by jagoll last updated on 25/Apr/20

	$t h a n k y o u s i r$
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