Problem 32 - 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 \(9 ?\)

Step-by-Step Solution

Answer

Only 324 is divisible by 9.

Step 1 - Understand the rule for divisibility by 9

A number is divisible by 9 if the sum of its digits is also divisible by 9.

Step 2 - Calculate sum of digits for each number

Break down each number into its individual digits and find the total sum.

Step 3 - Check sum of digits for divisibility by 9

Verify if the sum calculated in Step 2 for each number is divisible by 9.

Step 4 - Apply to each number

Calculate:1. 56: Sum of digits = 5 + 6 = 11 (Not divisible by 9)2. 200: Sum of digits = 2 + 0 + 0 = 2 (Not divisible by 9)3. 75: Sum of digits = 7 + 5 = 12 (Not divisible by 9)4. 35: Sum of digits = 3 + 5 = 8 (Not divisible by 9)5. 324: Sum of digits = 3 + 2 + 4 = 9 (Divisible by 9)6. 42: Sum of digits = 4 + 2 = 6 (Not divisible by 9)7. 812: Sum of digits = 8 + 1 + 2 = 11 (Not divisible by 9)8. 402: Sum of digits = 4 + 0 + 2 = 6 (Not divisible by 9)9. 784: Sum of digits = 7 + 8 + 4 = 19 (Not divisible by 9)10. 501: Sum of digits = 5 + 0 + 1 = 6 (Not divisible by 9)11. 2345: Sum of digits = 2 + 3 + 4 + 5 = 14 (Not divisible by 9)12. 111,111: Sum of digits = 1 + 1 + 1 + 1 + 1 + 1 = 6 (Not divisible by 9)13. 55,555: Sum of digits = 5 + 5 + 5 + 5 + 5 = 25 (Not divisible by 9)14. 3009: Sum of digits = 3 + 0 + 0 + 9 = 12 (Not divisible by 9)15. 2001: Sum of digits = 2 + 0 + 0 + 1 = 3 (Not divisible by 9)16. 1005: Sum of digits = 1 + 0 + 0 + 5 = 6 (Not divisible by 9)

Step 5 - List the results

From the previous step, only 324 is divisible by 9.

Key Concepts

divisibility by 9

Divisibility rules are shortcuts to help determine if one number can be divided by another without performing a full division. One useful rule is the rule for divisibility by 9.
To determine if a number is divisible by 9, here is a simple method:

• Add up all the digits in the number.
• If that sum is divisible by 9, then the original number is also divisible by 9.

This rule works because of the way our number system is structured. For instance, consider the number 324. By adding its digits (3 + 2 + 4), we get 9, which is divisible by 9. So, 324 is divisible by 9.

number properties

In basic mathematics, understanding number properties can make complex problems simpler. These properties include divisibility rules, such as those for 2, 3, and 9.
The property of divisibility by 9 shows that every number formed by rearranging the digits of a number divisible by 9 will also be divisible by 9. For example, if 324 is divisible by 9, then 423 or 243 will also be divisible by 9.
This consistency comes from the nature of our positional numeral system. Other properties to consider include:

• Even and odd numbers.
• Prime numbers.
• Multiples and factors.

These properties help in solving various mathematical problems efficiently.

basic mathematics

Learning the fundamentals of basic mathematics is essential. This includes operations like addition, subtraction, multiplication, and division. Knowledge of these basics helps in understanding more complex topics, such as divisibility rules.
For example, knowing how to add the digits of a number correctly is the first step in applying the rule for divisibility by 9. If we look at the numbers from the exercise:

• For 56, sum is 5 + 6 = 11 (not divisible by 9).
• For 200, sum is 2 + 0 + 0 = 2 (not divisible by 9).
• For 324, sum is 3 + 2 + 4 = 9 (divisible by 9).

Such exercises reinforce the importance of addition skills and understanding digit placement, which are foundational in any mathematical learning journey.