The sum of all the numbers formed

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

Solution:

 (c) We have, xyz + yzx + zxy

= (100x + 10y + z) + (100y + 10 z+ x) + (100z+ 10x + y) …(i)

= 100x + 10x + x + 10y + 100y + y + z+ 100z+ 10Z

= 111x + 111y + 111z = 111 (x + y + z)

= 3 x 37 x (x + y + z)

Hence, Eq. (i) is divisible by 37, but not divisible by 11,33 and 74.

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