Final answer:
The student's question requires identification of a number between 1 and 100 with a digit sum that is a prime and a square that is a multiple of 4. By following the given mathematical criteria, it is determined that no such number exists between 1 and 100 that fully satisfies these conditions.
Step-by-step explanation:
The question is about finding a number that meets certain mathematical conditions: it must be between 1 and 100, have digits that add up to a prime number, and its square must be a multiple of 4. To solve this problem, we can follow these steps:
Identify numbers within the given range whose squares are multiples of 4.
Find a number that satisfies both conditions.
First, note that the only single-digit prime numbers are 2, 3, 5, and 7 since the sum of the digits must be a prime number. Next, since the square of the number must be a multiple of 4, this implies the original number must be even.
Therefore, we can look for even numbers with digit sums of 2, 3, 5, or 7. By inspecting even numbers between 1 and 100 and checking their digit sum and whether their square is a multiple of 4, one can find that the number 46 fits all the criteria (4+6=10, which is not a prime number, but 4+6=10 and 1+0=1 which is also not a prime, thus the number does not fit the criteria completely, hence there is no number between 1 and 100 that satisfy the conditions).