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Gym Locker

Al had forgotten his locker combination and was going to be late for class

But he was able to remember the following things:

Each of the three numbers of the combination can be anywhere between 1 and 39 inclusive.
Each one was different and the sum of the first two numbers was 42. The last two numbers differed by only two, and all three numbers were primes.

His friend remembered the combination and told him, "If you take the three numbers of your combination and multiply them together, the last digit of the product will be....", and then whispered something in his ear.

With that, Al was able to deduce his combination and open his locker.

What was his combination?
Answer

Here are all the combinations which add up to 42:
5 + 37 = 42
11 + 31 = 42
13 + 29 = 42
23 + 19 = 42
29 + 13 = 42
31 + 11 = 42
37 + 5 = 42
41 + 1 = 42

And here are the permutations of that list where the last number differs by 2 from the penultimate number:
05 37 35
05 37 39
11 31 29
11 31 33
13 29 27
13 29 31
23 19 17
23 19 21
29 13 11
29 13 15
31 11 09
31 11 13
37 05 03
37 05 07
41 01 03

If you eliminate the non-primes, and examine their products (the last column), you get the narrower list:
11 31 29 --> 9
13 29 31 --> 7
23 19 17 --> 9
29 13 11 --> 7
31 11 13 --> 3
37 05 03 --> 5
37 05 07 --> 5
Although Al's friend's words were lost to history, it is clear that the product was '3', because any other answer wouldn't have been sufficient.

So the combination must have been 31-11-13.

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This page revised   August 3, 2004
 
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