Think of a three-digit number, then form a six-digit number by repeating its three digits (469 becomes 469,469).

arithmetic/digits/abcabc New

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Problem

Think of a three-digit number, then form a six-digit number by repeating
its three digits (469 becomes 469,469).  What is the probability that the
new number is divisible by 13?

Show solution

Certainty. It's also divisible by 7 and 11. Repeating the three
digits is equivalent to multiplying the number by 1,001, which
is 7 * 11 * 13.

-- Jonathan Haas <positron@primenet.com>

Any number xyz (where x, y, and z are digits) is then converted to
xyzxyz.

Ok, any number xyzxyz = xyz * 1001, which = xyz * 13 * 77,
so it would HAVE to be divisible by 13.

-- Fletch <bjames@skygames.com>