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Question 83332: I need to find the combination to a cylindrical lock with digits from 0-9 printed on each of the cylinders. Use these clues to open the lock:
-the lock has three cylinders
-the product of the tens digit and the ones digit is 8
-the three digit number is greater than 256
-the tens digit is the sum of the hundreds digit and the ones digit
What is the three-digit combination?
Thank you for your help...I'm desperate!
Answer by bucky(2189)   (Show Source):
You can put this solution on YOUR website! You can deduce this in a few steps.
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Call the three digits H for hundreds, T for tens, and U for units or ones.
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The first clue (#1) is that you are told that T times U equals 8. That means that T and U are
factors of 8. This leads to four possible combinations:
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T = 1 and U = 8
T = 2 and U = 4
T = 4 and U = 2
T = 8 and U = 1
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Second (#2) you are told that HTU is greater than 256. This tells you that H must be at least 2.
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Third (#3) you are told that T = H + U. Since you know from #2 that H is 2 or greater,
and from #1 that U is at least 1, then you know that H + U must be at least 3. Therefore,
you can say that T must be at least 3. Return to #1. From the four possible combinations,
T could be 1 or 2. But you just eliminated that possibility using #3. So now you know
that the possible combinations of T and U have been reduced to:
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T = 4 and U = 2
T = 8 and U = 1
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So the possible combinations of T and U are 42 and 81.
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Return to #3 which told you that T = H + U and solve for H by subtracting U from both sides
to get T - U = H. You know that T - U must be either 4 - 2 or 8 - 1. If it is 4 minus 2
Then H = 4 - 2 which makes H = 2. In this case HTU becomes 242. But #2 said that HTU
must be greater that 256. This leads to the conclusion that the only possible combination
for T and U that remains in the running is 81. And since H = T - U, then H = 8 - 1 = 7.
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You're there! H = 7, T = 8, and U = 1 for the combination 781.
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Check ...
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#1 Does T*U = 8? Yes 8*1 does equal 8.
#2 Is HTU greater than 256? Yes, 781 is greater than 256.
#3 Does T = H + U? Yes 8 does equal 7 + 1.
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The answer of 781 meets all the criteria that were given in the problem and is a correct
solution. Hope this helps you to see your way through the problem. Cheers!
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