Let $99$ numbers be $a_1,\dots ,a_{99}$.If not all the cards contain the same number,WLOG $a_1$ and $a_2$ are different from each other in $\mod{100}$.We consider the following $100$ numbers.
$a_1$
$a_2$
$a_1+a_2$
$\dots$
$a_1+\dots +a_{99}$
From the condition,these are $1$~$99$ in $\mod{100}$.By PHP,there are $2$ numbers which are congruent to each other in $\mod{100}$.These are not $(a_1,a_2)$.Thus difference between $2$ numbers is $0 \mod{100}$.This is contradiction.Therefore all the cards contain the same number.$\blacksquare$