Problem 35 - Fundamentals of Mathematics

Question

Specify which of the digits \(2,3,\) or 4 divide into \(9,462 .\)

Step-by-Step Solution

Answer

9,462 is divisible by 2 and 3, but not by 4.

Step 1: Check divisibility by 2

To check if a number is divisible by 2, we need to examine the last digit of the number. If the last digit is even (0, 2, 4, 6, 8), the number is divisible by 2. In this case, the last digit of 9,462 is 2, which is even, so 9,462 is divisible by 2.

Step 2: Check divisibility by 3

A number is divisible by 3 if the sum of its digits is divisible by 3. First, we calculate the sum of the digits of 9,462:\[9 + 4 + 6 + 2 = 21.\] Next, check if 21 is divisible by 3. Since 21 divided by 3 equals 7, which is an integer, the sum is divisible by 3. Hence, 9,462 is divisible by 3.

Step 3: Check divisibility by 4

For a number to be divisible by 4, the last two digits of the number must form a number that is divisible by 4. The last two digits of 9,462 are 62. Now, check if 62 is divisible by 4. Since 62 divided by 4 gives 15.5, which is not an integer, 62 is not divisible by 4. Thus, 9,462 is not divisible by 4.

Key Concepts

Divisibility by 2

Determining if a number is divisible by 2 is one of the simplest divisibility rules to remember. The rule requires you to look at the last digit of the number. If the final digit is 0, 2, 4, 6, or 8, the entire number is divisible by 2.
For instance, take the number 9,462. The last digit here is 2, which is an even number. This meets the requirement for divisibility by 2.
Thus, 9,462 passes the test and is divisible by 2. This rule makes it quick to check even very large numbers without needing to perform division.

• Last digit is even - number is divisible by 2.
• 9,462 ends in 2, hence it's divisible by 2.

Divisibility by 3

To determine if a number is divisible by 3, you need to sum all the digits of the number. When the total sum of these digits is divisible by 3, then the original number is also divisible by 3.
For example, considering the number 9,462, you begin by adding each digit: 9 + 4 + 6 + 2 = 21.
Now, you need to check if 21 can be divided by 3 without leaving a remainder.
21 divided by 3 yields 7, which is an integer, confirming that 9,462 is divisible by 3.
This approach simplifies the process, especially for numbers with multiple digits.

• Add the digits: 9 + 4 + 6 + 2 = 21.
• If the sum, 21, is divisible by 3, so is the original number, 9,462.

Divisibility by 4

Divisibility by 4 involves looking at the last two digits of a number. If these two digits form a number that is divisible by 4, then the entire number is divisible by 4.
In the case of 9,462, you examine the last two digits, which are 62.
Next, you determine whether 62 can be divided by 4 without leaving a remainder.
Divide 62 by 4, which gives 15.5, not an integer. Therefore, 9,462 is not divisible by 4.
This quick check can save time, especially when working with large numbers, making it a valuable method.

• Consider the last two digits: 62.
• If 62 is not divisible by 4, neither is 9,462.