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N is a 3-digit positive integer. The sum of all 3 digits of

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by Max@Math Revolution » Thu Dec 12, 2019 11:52 pm
[GMAT math practice question]

N is a 3-digit positive integer. The sum of all 3 digits of N is 17, and the sum of its hundreds digit and its units digit is 13. The new number, which is made by exchanging the digits in the hundreds position and the units digit,00000 is 99 less than the original number N. What is the value of N?

A. 944
B. 449
C. 548
D. 845
E. 746
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by henilshaht » Fri Dec 13, 2019 5:33 am
Instead of writing the equations, I find the following approach to be time-saving:

We know that if we exchange the hundred and the unit digit, the new number is 99 less than the original. With this info, let us try the options:

A. 944
New number: 449
944-449 > 99

B. 449
New number: 944
449-944 < 99

C. 548
New number: 845
548-845 < 99

D. 845
New number: 845
845-548> 99

E. 746
New number: 647
746-647 = 99
E is the answer.
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by Max@Math Revolution » Sun Dec 15, 2019 5:02 pm
=>

Assume N is a three-digit integer represented by xyz.
Then we have an algebraic expression N = 100x + 10y + z.
We have x + y + z = 17, x + z = 13 and 100z + 10y + x = 100x + 10y + z - 99.
If we subtract the first 2 equations we get (x + y + z) - (x + z) = 17 - 13 and y = 4.
When we substitute this into the last equation, we have 100z + 10*4 + x = 100x +10*4 + z - 99, 99z - 99x = -99. If we multiply by -1 we get 99x - 99z = 99 or x - z = 1.
Then we have x = 7 and z = 6, since x + z = 13 and x - z = 1.
So, the original number N is 746.

Therefore, E is the answer.
Answer: E
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