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Number System

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Number System

by adthedaddy » Thu Sep 24, 2015 7:12 pm
If two 2-digit positive integers have their resp tens digits exchanged, the difference between the pair of integers changes by 4. What is the greatest possible difference between the original pair of integers?
A) 76
B) 80
C) 82
D) 90
E) 94
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by Matt@VeritasPrep » Fri Sep 25, 2015 12:30 am
Let's call the first number ab and the second cd. Let's also assume a > c.

Any two-digit number ab can be written as 10a + b, so we have

(10a + b) - (10c + d) = x, where x is the original difference.

After exchanging the tens digits, we have

(10a + d) - (10c + b) = x - 4

Subtracting the second equation from the first gives us

10(a - c) + (b - d) - (10(a - c) + (d - b)) = 4

or (b - d) = 2

So the units digits of the original numbers must differ by 2.

To make the tens digits as different as possible, we could use a = 9 and c = 1. b and d must be such that b = d + 2, so we could choose b = 9 and d = 7. This gives us a difference of 99 - 17, or 82, and we're done. (We could also note that a difference ≥ 90 is impossible anyway, since we're dealing with two two-digit integers.)

Great question!
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by adthedaddy » Fri Sep 25, 2015 9:30 pm
Hi Matt,

Please help understand the following -
Any two-digit number ab can be written as 10a + b, so we have

(10a + b) - (10c + d) = x, where x is the original difference.

After exchanging the tens digits, we have

(10a + d) - (10c + b) = x - 4
Doesn't exchanging the tens digits mean we interchange 'a' and 'c' in the original equation. I cannot understand why you have interchanged 'b' and 'd' digits.
"Your time is limited, so don't waste it living someone else's life. Don't be trapped by dogma - which is living with the results of other people's thinking. Don't let the noise of others' opinions drown out your own inner voice. And most important, have the courage to follow your heart and intuition. They somehow already know what you truly want to become. Everything else is secondary" - Steve Jobs
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