If r is the remainder when ab is divided by ba, where both ab and ba

Archit3110 - Are these ab & ba pairs ? If so, none of the pairs are correctly identified.

For example in the "16 ; 61" pair if the divisor is 16, how can the remainder be 45 ?

i missed on the part a>b

gmatophobia

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