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09 Nov 2023, 11:31
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If \(r\) is the remainder when \(ab\) is divided by \(ba\), where \(ab\) and \(ba\) are positive two-digit integers with \(a > b\), what is the maximum possible value of \(r\)?
A. 9
B. 18
C. 27
D. 36
E. 45
A. 9
B. 18
C. 27
D. 36
E. 45
09 Nov 2023, 11:31
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Official Solution:
If \(r\) is the remainder when \(ab\) is divided by \(ba\), where \(ab\) and \(ba\) are positive two-digit integers with \(a > b\), what is the maximum possible value of \(r\)?
A. 9
B. 18
C. 27
D. 36
E. 45
Firstly, observe that \(a > b\) implies that \(ab > ba\).
\(ab\) dividing by \(ba\) yielding the remainder of \(r\) can be expressed as: \(ab = ba*q + r\).
To obtain the maximum remainder, we should maximize the difference between \(ab\) and \(ba\), ensuring that the quotient (\(q\)) is 1. In this case, the remainder (\(r\)) would be equal to the difference between \(ab\) and \(ba\).
For example, if \(ab = 98\) and \(ba = 89\), then the remainder is 9, with \(q=1\) and \(r\) being the difference between \(ab\) and \(ba\).
Similarly, if \(ab = 97\) and \(ba = 79\), the remainder is 18, where \(q=1\) and \(r\) is the difference between \(ab\) and \(ba\).
When \(ab = 96\) and \(ba = 69\), the remainder is 27, again with \(q=1\) and \(r\) as the difference between \(ab\) and \(ba\).
If \(ab = 95\) and \(ba = 59\), the remainder is 36, where \(q=1\) and \(r\) is the difference between \(ab\) and \(ba\).
If \(ab = 94\) and \(ba = 49\), the remainder is 45, with \(q=1\) and \(r\) being the difference between \(ab\) and \(ba\).
However, if \(ab = 93\) and \(ba = 39\), the remainder is 15. Here, \(q=2\), hence \(r\) is not the difference between \(ab\) and \(ba\), resulting in a smaller remainder.
Thus, the largest possible remainder is 45.
Answer: E
If \(r\) is the remainder when \(ab\) is divided by \(ba\), where \(ab\) and \(ba\) are positive two-digit integers with \(a > b\), what is the maximum possible value of \(r\)?
A. 9
B. 18
C. 27
D. 36
E. 45
Firstly, observe that \(a > b\) implies that \(ab > ba\).
\(ab\) dividing by \(ba\) yielding the remainder of \(r\) can be expressed as: \(ab = ba*q + r\).
To obtain the maximum remainder, we should maximize the difference between \(ab\) and \(ba\), ensuring that the quotient (\(q\)) is 1. In this case, the remainder (\(r\)) would be equal to the difference between \(ab\) and \(ba\).
For example, if \(ab = 98\) and \(ba = 89\), then the remainder is 9, with \(q=1\) and \(r\) being the difference between \(ab\) and \(ba\).
Similarly, if \(ab = 97\) and \(ba = 79\), the remainder is 18, where \(q=1\) and \(r\) is the difference between \(ab\) and \(ba\).
When \(ab = 96\) and \(ba = 69\), the remainder is 27, again with \(q=1\) and \(r\) as the difference between \(ab\) and \(ba\).
If \(ab = 95\) and \(ba = 59\), the remainder is 36, where \(q=1\) and \(r\) is the difference between \(ab\) and \(ba\).
If \(ab = 94\) and \(ba = 49\), the remainder is 45, with \(q=1\) and \(r\) being the difference between \(ab\) and \(ba\).
However, if \(ab = 93\) and \(ba = 39\), the remainder is 15. Here, \(q=2\), hence \(r\) is not the difference between \(ab\) and \(ba\), resulting in a smaller remainder.
Thus, the largest possible remainder is 45.
Answer: E
25 Jun 2024, 20:27
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Thank you for the question Bunuel
Here is how I tried it
To obtain the maximum remainder, the quotient q should be 1. Putting q=1 in the equation
ab = 1*ba + R
Now ab = 10A+B and ba= 10B + A .
Therefore equation becomes
10A + B = 10B +A +R
9(A-B) = R
A-B = R/9
Now we need a value of R which is divisible by 9. From the options, we can see the largest value is 45
Hence answer is E
Here is how I tried it
To obtain the maximum remainder, the quotient q should be 1. Putting q=1 in the equation
ab = 1*ba + R
Now ab = 10A+B and ba= 10B + A .
Therefore equation becomes
10A + B = 10B +A +R
9(A-B) = R
A-B = R/9
Now we need a value of R which is divisible by 9. From the options, we can see the largest value is 45
Hence answer is E
