6th Grade Math Textbook, Fundamentals

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Problem 2:The four digits 1, 2, 3, and 4 can be used to form exactly 24 different
four-digit numbers. Of these, how many are evenly divisible by 4?

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Lesson 10-13 for exercise sets.

Read to understand what is being asked.

List the facts and restate the question.

Facts: 24 four-digit numbers can be formed from 1, 2, 3, and 4.

Question:How many of these 24 numbers are divisible by 4?

Select a strategy.

Because there are only 24 such numbers, you could attempt to

consider all the possibilities.

Apply the strategy.

Recall the divisibility criteria for 4: A number is divisible by 4 if and only

if the number formed by the last two digits of the number is divisible by 4.

For example, 520 is divisible by 4 because 20 is divisible by 4; 526 is not

divisible by 4 because 26 is not divisible by 4. Also, note that for any fixed

last two digits, there are two possible four-digit numbers. For example, if the

last two digits are 12, then the first two digits must be 34 or 43. So the two

possible numbers are 3412 and 4312.

Using these ideas, you can sort through the possibilities pretty quickly.

• There are six numbers ending in 1: two each with last two digits 21, 31, and 41.
  None of these is divisible by 4.


• There are six numbers ending in 2: two each with last digits 12, 32, 42.
  The two with endings 12 (3412 and 4312) and 32 (1432 and 4132) are
  divisible by 4. So far, this makes 4 numbers that are divisible by 4.


• There are six numbers ending in 3: two each with last digits 13, 23, 43.
  None of these is divisible by 4.


• There are six numbers ending in 4: two each with last digits 14, 24, 34.
  Only the two with ending 24 (1324 and 3124) are divisible by 4. This adds
  2 more to the list of numbers divisible by 4.
  In all, then, there are only 6 of these numbers that are divisible by 4.

Check to make sure your answer makes sense.

A complete list of the 24 different numbers shows that the solution is correct.

1234, 1243, 1324, 1342, 1423, 1432,

2134, 2143, 2314, 2341, 2413, 2431,

3124, 3142, 3214, 3241, 3412, 3421,

4123, 4132, 4213, 4231, 4312, 4321

Only the six underlined numbers are divisible by 4.

Check to make sure your answer makes sense.

You could cut thin paper strips of these eight lengths and then form

the triangles listed in the table. In this process, it would also be clear

why certain combinations (such as 1-2-3 or 3-4-8) do notform triangles.