**Possible Remainders When N is Divided by 12**

To find the possible remainders when N is divided by 12, we need to consider the given information that N2 leaves a remainder of 1 when divided by 24. Let's break down the problem into smaller steps.

**Step 1: Understand the Problem**
The problem states that N2 (N squared) leaves a remainder of 1 when divided by 24. This means that N2 is congruent to 1 modulo 24, or in other words, N2 ≡ 1 (mod 24).

**Step 2: Find the Possible Values for N**
To find the possible values for N, we need to consider the quadratic residues modulo 24. A quadratic residue is an integer that can be obtained by squaring another integer.

We can start by listing the quadratic residues modulo 24:
0, 1, 4, 9, 16, 5, 20, 17, 13, 21, 18, 10.

Since N2 ≡ 1 (mod 24), N can be any integer that satisfies N2 ≡ 1 (mod 24). This means that N can take the values of the square roots of 1 modulo 24.

**Step 3: Find the Square Roots of 1 Modulo 24**
To find the square roots of 1 modulo 24, we can use the concept of modular arithmetic. We need to find the integers x that satisfy x2 ≡ 1 (mod 24).

By trying out different values, we can find that the square roots of 1 modulo 24 are ±1, ±5, ±7, ±11, ±13, ±17, ±19, and ±23.

**Step 4: Find the Possible Remainders When N is Divided by 12**
Now that we have found the possible values for N, we can find the possible remainders when N is divided by 12.

When N = ±1, ±5, ±7, ±11, ±13, ±17, ±19, or ±23, we can divide N by 12 and find the remainders.

For example, when N = 1:
1 ÷ 12 = 0 remainder 1.

Similarly, when N = 5:
5 ÷ 12 = 0 remainder 5.

Continuing this process for all the possible values of N, we can find the possible remainders when N is divided by 12.

**Possible Remainders When N is Divided by 12:**
0, 1, 5, 7, 11.

Therefore, the possible remainders we can get when N is divided by 12 are 0, 1, 5, 7, and 11.

The possible numbers of n are 5,7 so the possible remainders when these numbers are divided by 12 is 5,7.