Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question

4 Answers 4

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 $4$.

share|improve this answer

Hint:

$xy+ab+cd+ef=221$

$9(x+a+c+e)+y+b+d+f+(x+a+c+e)=221$

$\equiv(x+a+c+e)+y+b+d+f \equiv 5 \mod 9 $

Now you know which number has to be excluded.

share|improve this answer

Hint $\ $ Adding in the number $\rm\,0\,d\,$ formed by the missing digits $\rm\,0,d\,$ and casting nines shows

$\rm\qquad\quad mod\ 9\!:\ \ 2\!+\!2\!+\!1\!+\!d\, \equiv\, 0\!+\!1\!+\!2\!+\cdots + 9\,\equiv\, 0,\ $ i.e. $\rm\ 5+d\equiv 0\ \Rightarrow\ d\equiv -5\equiv 4$

share|improve this answer

Suppose the numbers are $a_1b_1, \ a_2b_2, \ a_3b_3, \ a_4b_4.$

Then from $$\begin{array}{cc} a_1b_1& \\ a_2b_2& \\a_3b_3& \\ a_4b_4 &+ \\ \hline \\221 \end{array} $$ we conclude that only three possibilities can happen.
Either
$b_1+b_2+b_3+b_4=31$ and $a_1+a_2+a_3+a_4=19\,,$
$b_1+b_2+b_3+b_4=21$ and $a_1+a_2+a_3+a_4=20$ or
$b_1+b_2+b_3+b_4=11$ and $a_1+a_2+a_3+a_4=21\,.$
Since $a_1,\ a_2,\ a_3,\ a_4,\ b_1,\ b_2,\ b_3,\ b_4$ are all different and $1+2+\ldots+9=45$ we conclude that only the second possibility can happen and $4$ is not included in the digits $a_i,b_i$ ($a_1+a_2+a_3+a_4+b_1+b_2+b_3+b_4=41$).

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.