Math Problem Statement

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?

Solution

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=56364312 + 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!

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Math Problem Analysis

Mathematical Concepts

Divisibility Rules
Permutations

Formulas

Divisibility by 4: A number is divisible by 4 if its last two digits form a number divisible by 4

Theorems

Permutation principles for arranging digits

Suitable Grade Level

Grades 8-10