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## Solution to "Three-digit numbers"

May 1998

When we write a three-digit number in base-ten we are really saying "so many hundreds", "so many tens" and "so many units". If our number is *n* and the digits are represented by *x*, *y* and *z* (reading from left to right) then we can write:

*n* = 100*x* + 10*y* + *z*

We are told that

*y* = *x* + *z*

Therefore

*n* = 100*x* + 10(*x* + *z*) + *z*

*n* = 110*x* + 11*z*

*n* = 11(10*x* + *z*)

therefore *n* must be divisible by 11!