Floor function equation

Algebra Level 2

5 x = 15 \big\lfloor 5 - \lfloor x \rfloor \big\rfloor = 15

If all the values of x x that satisfy the equation above are in the interval a x < b a \le x < b , find the product a b ab .


Note: x \lfloor x \rfloor is the floor function, or the greatest integer function.


The answer is 90.

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3 solutions

Anthony Muleta
Nov 19, 2015

In order for the equation to be true, 5 x 5 - \left\lfloor x \right\rfloor must be greater than or equal to 15 15 and less than but not equal to 16 16 i.e.

15 5 x < 16 15 \le 5 - \left\lfloor x \right\rfloor < 16

10 x < 11 10 \le - \left\lfloor x \right\rfloor < 11

11 < x 10 -11 < \left\lfloor x \right\rfloor \le -10 .

Since x \left\lfloor x \right\rfloor is an integer, x = 10 \left\lfloor x \right\rfloor = -10 , as this is the only integer in the interval ( 11 , 10 ] (-11,-10] .

x x must therefore be greater than or equal to 10 -10 and less than but not equal to 9 -9 i.e. 10 x < 9 -10 \le x < -9 .

a = 10 a=-10 , b = 9 b=-9 , a b = 90 ab=\boxed {90} .

amzng........

Tanmay Nayek - 2 years, 2 months ago

very well said!

Krish Shah - 1 year, 2 months ago

Let y y = = x \left\lfloor x \right\rfloor

As y y is an integer so
5 y \left\lfloor 5 - y \right\rfloor = = 5 y 5 - y = = 10 10 \Rightarrow y = 10 y = -10

For any 10 x < 9 -10 \le x \lt -9 , , x \left\lfloor x \right\rfloor = = 10 -10

So here ( a = 10 , b = 9 ) (a = -10,b = -9) a b = ( 10 ) ( 9 ) = 90 ab = (-10)(-9) = \boxed{90}

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