A 2-digit number is reversed. The larger of the two numbers is divided by the smaller one. What is the largest possible remainder?

Correct Option (D)

Let the 2-digit number be represented as $10a+b$, where $a$ is the tens digit and $b$ is the units digit. For both the original number and its reverse to be valid 2-digit numbers, both $a$ and $b$ must be integers from 1 to 9 (i.e., $a,b\in \{1,2,...,9\}$).

The reversed number will be $10b+a$.

The problem requires dividing the larger of these two numbers by the smaller one. Let $D$ be the smaller number (divisor) and $R$ be the remainder. According to the properties of division, the remainder $R$ must satisfy .

To identify the largest possible remainder, we should consider scenarios where the divisor $D$ is relatively large, allowing for a larger potential remainder $R$ that is just below $D$. Given the options, the maximum possible remainder is likely to be 45. If the remainder is 45, the divisor $D$ must be at least 46.

We systematically examine pairs of numbers $(10a+b,10b+a)$ where the smaller number (divisor $D$) is 46 or greater:

• Consider the number 49 ($a=4,b=9$). Its reverse is 94.
  • The smaller number is 49 (divisor).
  • The larger number is 94 (dividend).
  • Dividing 94 by 49 yields: $94=1\times 49+45$. The remainder is 45.

Further examination of other relevant pairs confirms that 45 is the maximum remainder:

• For numbers 46 ($a=4,b=6$) and 64 ($10b+a$): $64÷46⟹R=18$.
• For numbers 47 ($a=4,b=7$) and 74 ($10b+a$): $74÷47⟹R=27$.
• For numbers 48 ($a=4,b=8$) and 84 ($10b+a$): $84÷48⟹R=36$.
• For numbers 59 ($a=5,b=9$) and 95 ($10b+a$): $95÷59⟹R=36$.
• For numbers 69 ($a=6,b=9$) and 96 ($10b+a$): $96÷69⟹R=27$.
• For numbers 79 ($a=7,b=9$) and 97 ($10b+a$): $97÷79⟹R=18$.
• For numbers 89 ($a=8,b=9$) and 98 ($10b+a$): $98÷89⟹R=9$.

The largest remainder obtained through this systematic analysis is 45.

Incorrect Options:

• Option A (9): A remainder of 9 is a possible outcome (e.g., 98 divided by 89 yields a remainder of 9). However, this is not the largest possible remainder.

• Option B (27): A remainder of 27 is a possible outcome (e.g., 96 divided by 69 yields a remainder of 27). This is not the largest possible remainder.

• Option C (36): A remainder of 36 is a possible outcome (e.g., 95 divided by 59 yields a remainder of 36). This is not the largest possible remainder.