For the numbers 653, 749, 872, 933, and 541, if in each number the first digit is replaced by the third digit, the second digit is replaced by the first digit, and the third digit is replaced by the second digit, which resulting number is the greatest?

Introduction / Context:
This problem tests your ability to manipulate digits of three digit numbers according to a specific rule and then compare the transformed results. Such digit manipulation and comparison tasks are common in arithmetic reasoning and help assess attention to detail as well as understanding of place value.

Given Data / Assumptions:
- The original numbers are 653, 749, 872, 933, and 541.- For each number, a digit transformation rule is applied.- The rule is: first digit is replaced by the third digit, second digit is replaced by the first digit, and third digit is replaced by the second digit.- We then compare the resulting transformed numbers to find the largest.

Concept / Approach:
Let a generic three digit number have digits A (hundreds), B (tens), and C (units). The rule says that A is replaced by C, B is replaced by A, and C is replaced by B. In other words, the new number has digits C, A, B in that order. Therefore, for any given three digit number, we can rewrite it in this permuted form and then evaluate the resulting numbers. After computing all the new numbers, we simply compare them numerically to find the greatest one.

Step-by-Step Solution:
- Take 653: here A = 6, B = 5, C = 3. The new number is C A B, that is 3 6 5 or 365.- Take 749: A = 7, B = 4, C = 9. The new number is 9 7 4, which is 974.- Take 872: A = 8, B = 7, C = 2. The new number is 2 8 7, which is 287.- Take 933: A = 9, B = 3, C = 3. The new number is 3 9 3, which is 393.- Take 541: A = 5, B = 4, C = 1. The new number is 1 5 4, which is 154.- Now compare the transformed values: 365, 974, 287, 393, and 154.- Among these, 974 is the largest.- 974 came from the original number 749.

Verification / Alternative check:
One quick check is to observe that the new hundreds digit is the original units digit. So any original number with the largest units digit is a strong candidate to give the largest transformed number. Among the units digits 3, 9, 2, 3, and 1, the largest is 9 from 749. That means the transformed number from 749 will have 9 in the hundreds place and is likely to be the greatest. The full numerical calculation confirms this intuition.

Why Other Options Are Wrong:
- 933 gives 393 after transformation, which is far smaller than 974.- 872 becomes 287, which has hundreds digit 2 and is clearly smaller.- 653 becomes 365, still below 974.- 541 becomes 154, which is the smallest among the transformed numbers.

Common Pitfalls:
A common mistake is to misapply the digit swapping rule, for example exchanging only two digits or misreading the order C, A, B. Another issue is forgetting that the largest hundreds digit dominates the comparison of three digit numbers. Always carefully write down the mapping from old to new digits and compute each transformed number before comparing them. Doing quick checks on place values helps verify that your answers are reasonable.

Final Answer:
The original number that becomes the greatest after the given transformation is 749.