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

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

The problem requires dividing the larger of these two numbers by the smaller one. Let $D$D be the smaller number (divisor) and $R$R be the remainder. According to the properties of division, the remainder $R$R must satisfy $0 \le R < D$0≤R<D.

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

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

• Consider the number 49 ($a=4, b=9$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$94=1×49+45. The remainder is 45.

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

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