Ollie arranges the numbers 1 to 6 to form three 2-digit numbers.
Exactly one of his numbers divides by 3 and exactly one other divides by 4. None of them divide by 5.
The three numbers add to 93.
What is the number which does not divide by either 3 or 4?

We are tasked with finding the number that neither divides by 3 nor by 4 among three 2-digit numbers formed from the digits 1 to 6, with certain conditions. The sum of the three numbers is 93, exactly one number divides by 3, exactly one other divides by 4, and none of them divides by 5.

### Step 1: Divisibility rules
- **Divisibility by 3**: A number is divisible by 3 if the sum of its digits is divisible by 3.
- **Divisibility by 4**: A number is divisible by 4 if the number formed by its last two digits is divisible by 4.
- **Divisibility by 5**: A number is divisible by 5 if it ends in 0 or 5. Since none of the numbers divides by 5, none of the numbers will end in 0 or 5.

### Step 2: Possible 2-digit numbers
The digits we are working with are 1, 2, 3, 4, 5, and 6. These digits must be arranged into three 2-digit numbers. The sum of these digits is:
\[
1 + 2 + 3 + 4 + 5 + 6 = 21
\]
This information will help us check our final answers.

### Step 3: Dividing the numbers
Let the three 2-digit numbers be \( N_1 \), \( N_2 \), and \( N_3 \). According to the problem:
- One of these numbers divides by 3.
- Another divides by 4.
- The third does not divide by either 3 or 4.

### Step 4: Strategy for forming numbers
We know that the total sum of \( N_1 + N_2 + N_3 = 93 \), and we will try to form numbers that satisfy the divisibility conditions.

### Step 5: Trial and error approach
We will form various combinations of numbers using the digits 1 to 6, ensuring that the sum of the numbers equals 93. After checking different possibilities, one possible valid solution is:

- \( N_1 = 36 \): divisible by 4 (since 36 is divisible by 4).
- \( N_2 = 21 \): divisible by 3 (since the sum of the digits, 2 + 1 = 3, is divisible by 3).
- \( N_3 = 36 \).

### Step 

Ollie arranges the numbers 1 to 6 to form three 2-digit numbers. Exactly one of his numbers divides by 3 and exactly one other divides by 4. None of them divide by 5. The three numbers add to 93. What is the number which does not divide by either 3 or 4?

Explore this challenging divisibility problem where you arrange the numbers 1 to 6 to form three 2-digit numbers. Learn how to apply divisibility rules to find the number that does not divide by either 3 or 4, while ensuring the sum is 93. Suitable for students in Grades 5-7.

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