Question

The sum of all the numbers formed by the digits $x,y$ and $z$ of the number $xyz$ is divisible by

• A. $11$
• B. $33$
• C. $37$
• D. $74$

Answer 1

The correct option is C

$37$

Step 1: Explanation for correct answer

Option (c): $37$

It is known that, three numbers which can be formed by the using the digits $x,y$ and $z$ are $xyz,yzx$ and $zxy$.

1. The general form of $xyz=100x+10y+z$
2. The general form of $yzx=100y+10z+x$
3. The general form of $zxy=100z+10x+y$

Add the three numbers, we will get

$xyz+yzx+zxy=(100x+10y+z)+(100y+10z+x)+(100z+10x+y)$

$xyz+yzx+zxy=100x+10x+x+100y+10y+y+100z+10z+z$

$xyz+yzx+zxy=111x+111y+111z$

Now, simplify the expression, we will get

$xyz+yzx+zxy=111(x+y+z)$

Now, it is clear that $111$ is the common factor. So,

$xyz+yzx+zxy$ is divisible by $111$.

Also $xyz+yzx+zxy$ will be divisible by factors of $111$.

$37$ is the factor of $111$.

Thus, option (c) is the correct answer.

Step 2: Explanation for incorrect answers

Option (a): $11$

Option (b): $33$

Option (d): $74$

From the given options, $11,33,$ and $74$ are not the factors of $111$ whereas $37$ is the factor of $111$.

Thus, option (a), (b) and (d) are incorrect answers.

Hence, option (c) is correct .