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number theory senior

Problem

The remainder can be defined for all real numbers

x

and

y

with

y \neq = 0

by

rem (x, y) = x - y ⌊ \frac{x}{y} ⌋

where

⌊ \frac{x}{y} ⌋

denotes the greatest integer less than or equal to

\frac{x}{y}

. What is the value of

rem (\frac{3}{8}, - \frac{2}{5})

?

(A)

- \frac{3}{8}

(B)

- \frac{1}{40}

(C)

0

(D)

\frac{3}{8}

Solution

The value, by definition, is

rem (\frac{3}{8}, - \frac{2}{5}) = \frac{3}{8} - (- \frac{2}{5}) ⌊ \frac{\frac{3}{8}}{- \frac{2}{5}} ⌋ = \frac{3}{8} - (- \frac{2}{5}) ⌊ \frac{3}{8} \times \frac{- 5}{2} ⌋ = \frac{3}{8} - (- \frac{2}{5}) ⌊ \frac{- 15}{16} ⌋ = \frac{3}{8} - (- \frac{2}{5}) (- 1) = \frac{3}{8} - \frac{2}{5} = - \frac{1}{40} .

Final answer

B