Question

Let $f$ be a function defined on the set of non-negative integers and taking values in the same set. Given that
i) $x-f\left(x\right)=19\left[\frac{x}{19}\right]-90\left[\frac{f\left(x\right)}{90}\right]\forall$ non-negative integers $x$. $\left[x\right]$ denotes greatest integer functions.
ii) $1900\le f\left(1990\right)\le 2000$. Then possible values of $f\left(1990\right)$ can take.

• A. $2004,2094$
• B. $1804,1994$
• C. $1904,1994$
• D. $1894$

Answer 1

The correct option is C $1904,1994$

1919x−19[x19]=90f(x)90−90[f(x)90]1919x−19[x19]=90f(x)90−90[f(x)90]

$\therefore korf\left(1990\right)canbe1904or1994.$