Question

let xyz be a three digit number ,where(x+y+z) is not a multiple of 3. then (xyz+yzx+zxy) is not divisible by 9,3,37,or(x+y+z)

Answer 1

Given:

Let xyz be a three digit number ,where(x+y+z) is not a multiple of 3.

To Find:

then (xyz+yzx+zxy) is not divisible by 9,3,37,or(x+y+z).

Solution:

xyz numbers is formed by the using the digits x, y and z  and other two numbers yzx and zxy also formed by using x, y and z.

here, it is given that- (x+ y+ z) is not a multiple of 3.

The general form of xyz = 100x + 10y + z.

The general form of yzx = 100y + 10z + x.

The general form of zxy = 100z + 10x + y.

Using above, add the three numbers, we get ;

$xyz + yzx + zxy = (100x + 10y + z) + (100y + 10z + x) + (100z + 10x + y)\\ = 100x + 10x + x + 100y + 10y + y + 100z + 10z + z\\ = 111x + 111y + 111z.$

Now, on 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.

from given options 9 is not a factor of 111 and 3, 37 are factor of 111.

but it is given that - (x+ y+ z) is not  multiple of 3

⇒ (xyz+ yzx+ zxy) is not divisible by 3, 9, 37 or (x+y+z).