Problem 31 - Basic College Mathematics

Question

To answer Exercises \(25-32,\) consider the following numbers. \(\begin{array}{rrrr}56 & 200 & 75 & 35 \\ 324 & 42 & 812 & 402 \\ 784 & 501 & 2345 & 111,111 \\ 55,555 & 3009 & 2001 & 1005\end{array}\) Which of the above are divisible by \(10 ?\)

Step-by-Step Solution

Answer

200

Step 1 - Identify the Divisibility Rule for 10

A number is divisible by 10 if its last digit is 0.

Step 2 - Review Each Number

Examine the last digit of each given number: 56 (last digit: 6), 200 (last digit: 0), 75 (last digit: 5), 35 (last digit: 5), 324 (last digit: 4), 42 (last digit: 2), 812 (last digit: 2), 402 (last digit: 2), 784 (last digit: 4), 501 (last digit: 1), 2345 (last digit: 5), 111,111 (last digit: 1), 55,555 (last digit: 5), 3009 (last digit: 9), 2001 (last digit: 1), 1005 (last digit: 5).

Step 3 - Identify Which Numbers End in 0

From the list, select the numbers that have a last digit of 0. These numbers are divisible by 10.

Key Concepts

divisibility by 10

Before we dive into the list of numbers, let's understand the rule for divisibility by 10. Divisibility rules help us determine if a number can be divided by another number without leaving a remainder. For a number to be divisible by 10, its last digit must be 0. This is because 10 is a multiple of the base-10 numeral system, which means only numbers ending in 0 can be split evenly by 10. So, whenever you see a number and you're asked if it's divisible by 10, just look at its last digit. If that digit is 0, then the number is divisible by 10.

last digit

The 'last digit' rule might sound simple, but it's incredibly powerful for quickly analyzing numbers. For the numbers given in the exercise: 56, 200, 75, 35, 324, 42, 812, 402, 784, 501, 2345, 111111, 55555, 3009, 2001, and 1005, let's inspect their last digits:

• 56: Last digit is 6
• 200: Last digit is 0 (Divisible by 10)
• 75: Last digit is 5
• 35: Last digit is 5
• 324: Last digit is 4
• 42: Last digit is 2
• 812: Last digit is 2
• 402: Last digit is 2
• 784: Last digit is 4
• 501: Last digit is 1
• 2345: Last digit is 5
• 111111: Last digit is 1
• 55555: Last digit is 5
• 3009: Last digit is 9
• 2001: Last digit is 1
• 1005: Last digit is 5

Only the number 200 ends with 0. Thus, from this list, only 200 is divisible by 10.

number properties

Understanding the properties of numbers is essential in math. When we talk about number properties in the context of divisibility, we are focusing on how certain features, like the last digit, can tell us broader things about the number. Knowing that a number ending in 0 is divisible by 10 can save a lot of calculation time. It's all about pattern recognition.

• **Consistency**: No matter what the number is, if the last digit is 0, it will always be divisible by 10.
• **Shortcut**: Instead of actually dividing each number by 10, just check the last digit. If it's not 0, you can immediately say it is not divisible by 10.

So, identifying number properties is about recognizing these patterns and using them to simplify calculations. It’s an important skill that will help you in higher-level math and everyday problem-solving.