The sum of four two digit numbers is $221$. None of the eight digits is 0 and none of them are same. Which of the following is not included among the eight digit ?
$$(a) \;\;1 \\ (b)\;\; 2 \\ (c)\;\; 3\\ (d)\;\; 4$$
Is there any shortcut to solve this question as I got the answer which is $(d)\;\; 4$ by trial and error method. Please suggest
Best Answer
It is enough to know what is the last number modulo $9$.
Since $10a+b \equiv a+b \pmod 9$, $221 \equiv 5 \pmod 9$, and $1+2+3+4+5+6+7+8+9 = 45 \equiv 0 \pmod 9$, if the unused digit is $x$, then the sum of all the used digits is $221 \equiv 5$, and it also is $45-x \equiv -x$.
So $x \equiv -5 \equiv 4 \pmod 9$, and therefore the unused digit is $4$.