https://file.aiquickdraw.com/mathbot/file/1729639356984tejix4sh.jpg

The problem asks for the sum of the largest and smallest four-digit multiples of 4 that can be formed using the digits 1, 2, 3, and 4 exactly once.

### Step-by-step approach:

1. **Condition for divisibility by 4**: A number is divisible by 4 if its last two digits form a number divisible by 4. So, when arranging the digits 1, 2, 3, and 4 to form four-digit numbers, we need to ensure that the last two digits create a number divisible by 4.

2. **Forming the largest multiple of 4**:
   - The largest possible four-digit number using the digits 1, 2, 3, and 4 is 4321. However, we need to check whether it is divisible by 4.
   - The last two digits are 21, which is not divisible by 4. So, 4321 is not divisible by 4.
   - The next largest number is 4312. The last two digits, 12, are divisible by 4. So, 4312 is the largest four-digit multiple of 4.

3. **Forming the smallest multiple of 4**:
   - The smallest possible four-digit number using the digits 1, 2, 3, and 4 is 1234. Checking the divisibility of the last two digits, 34 is not divisible by 4.
   - The next smallest number is 1243. The last two digits, 43, are not divisible by 4.
   - The next number is 1324. The last two digits, 24, are divisible by 4. Therefore, 1324 is the smallest four-digit multiple of 4.

4. **Sum of the largest and smallest multiples of 4**:
   - The largest multiple is 4312, and the smallest is 1324.
   - Sum: \( 4312 + 1324 = 5636 \).

### Final Answer:
The sum of the largest and smallest four-digit multiples of 4 is **5636**.

Would you like further details or have any other questions?

Here are some related questions:
1. How do you determine divisibility rules for numbers other than 4?
2. Can you form other multiples of 4 using the same digits?
3. What happens if you add another digit to the set?
4. How can divisibility by 8 or 16 be checked similarly?
5. What other mathematical properties can we explore from this set of numbers?

**Tip**: For divisibility by 4, always focus on the last two digits of the number to simplify your calculations!

What is the sum of the largest and smallest positive four-digit multiples of 4 that can be written using each of the digits 1, 2, 3, and 4 exactly once?

Learn how to find the sum of the largest and smallest four-digit multiples of 4 that can be created using the digits 1, 2, 3, and 4 exactly once. This problem involves understanding divisibility rules and arranging permutations. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation