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Re: If x, y, and z are integers is (x+y+z) a multiple of 9? [#permalink]
30 Sep 2013, 10:47
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If x, y, and z are digits is (x+y+z) a multiple of 9?
1) the two digit number yz is a multiple of 9 2) If xyz is a three digit number, 3(xyz-1)+3 is a multiple of 9
I'm happy to help with this.
First of all, if the sum of the digits of a number equals a multiple of 9, then the number itself is a multiple of 9. For example, 657 has a sum of digits 6 + 5 + 7 = 18, which is a multiple of 9, so this automatically means 657 is a multiple of 9. See http://magoosh.com/gmat/2012/gmat-divis ... shortcuts/
Statement #1: the two digit number yz is a multiple of 9 This gives us zero information about x, so we cannot determine an answer. The three digit number xyz could be 645 or 945 --- the latter is divisible by 9, but the former isn't. This statement, alone and by itself, is insufficient.
Statement #2: If xyz is a three digit number, 3(xyz-1)+3 is a multiple of 9 First of all, 3(xyz-1)+3 = 3*xyz - 3 + 3 = 3*xyz. If 3*xyz is a multiple of 9, then all this means is that xyz must be a multiple of 3. Thus, either 645 or 945 would work. This statement, alone and by itself, is insufficient.
Combined statements: Notice that the two numbers 645 and 945 each satisfy both statements, but the former is not a multiple of 9, and the latter is. Thus, even with both statements, we can still come up with different numbers that yield different answer to the prompt question. Thus, even together, the combined statements are insufficient.