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Delicate Divisibility (Posted on 2010-10-19) Difficulty: 3 of 5
For a positive base ten integer of the form ABCD drawn at random between 1111 and 9999 inclusively, determine the probability that ABCD is divisible by each of AB, AC, AD, BD and CD; but, not divisible by BC.

Note: Each of the letters in bold represents a digit from 1 to 9, whether same or different.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Solution ABACADABrA (Solution) | Comment 1 of 7
For ABCD to be divisible by CD, AB must be divisible by CD.
For ABCD to be divisible by AB, CD must be divisible by AB.

This means AB = CD to rewrite ABCD as ABAB.
ABAB = 101*AB
AC=AA so ABAB must be divisible by 11.
which means AB must be divisible by 11.

but this makes A=B
which makes the number divisible by BC.

(So any ABCD divisible by AB, CD and AC is also divisible by BC)

The probability of drawing a number that does not exist is zero.

  Posted by Jer on 2010-10-19 17:07:14
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