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Chapter 1: Problem 65

Which of the following numbers is not divisible by 9? $$216, 783, 594, 928 $$

Short Answer

Expert verified
928 is not divisible by 9.

Step by step solution

01

Understand Divisibility Rule for 9

A number is divisible by 9 if the sum of its digits is a multiple of 9.
02

Calculate Sum of Digits for Each Number

First, calculate the sum of the digits for each number: - For 216: \(2 + 1 + 6 = 9\)- For 783: \(7 + 8 + 3 = 18\)- For 594: \(5 + 9 + 4 = 18\)- For 928: \(9 + 2 + 8 = 19\)
03

Check Each Sum Against Divisibility by 9

Check if the sums calculated are multiples of 9: - 9 is a multiple of 9. - 18 is a multiple of 9. - 18 is a multiple of 9. - 19 is not a multiple of 9.
04

Identify the Number That Is Not Divisible by 9

Based on the sums, identify which number is not divisible by 9: 928 has a sum of digits equal to 19, which is not divisible by 9.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Divisibility Rules
Divisibility rules help us quickly determine whether a number is divisible by another number without doing complex calculations. For example, the rule for divisibility by 9 states that a number is divisible by 9 if the sum of its digits is also divisible by 9. This is an efficient shortcut, especially when dealing with large numbers, because calculating the sum of digits is much easier than performing long division. Applying this rule saves time and effort. Remembering key divisibility rules, such as those for 2, 3, 5, and 9, can simplify many math problems considerably. They act as tools in your math toolkit, making problem-solving more accessible.
Sum of Digits
The sum of digits is simply the sum of all the individual digits that make up a number. For instance, in the number 216, the digits are 2, 1, and 6. We add these together to get 9, i.e., \(2 + 1 + 6 = 9\). Calculating the sum of digits is the first step in applying many divisibility rules, such as those for 9 and 3. It turns a potentially complicated division problem into a simple addition problem.To practice, take any number, separate its digits, and add them together. This practice helps to apply divisibility rules efficiently and quickly checks if a number meets the criteria for divisibility without performing division.
Multiples
Multiples are the result of multiplying a number by an integer. For example, the multiples of 9 include 9, 18, 27, and so on. Recognizing multiples is integral to understanding divisibility, since if the sum of a number鈥檚 digits is a multiple of 9, the original number is divisible by 9. This concept is at the heart of checking divisibility. Let's look at it this way:
  • If a number's digit sum is 9, it is a multiple of 9.
  • The number 18 is also a multiple of 9 because 9 times 2 equals 18.
  • A digit sum of 19, however, is not a multiple of 9, thus not divisible by 9.
Being familiar with multiples thus reinforces our understanding of divisibility and supports our number sense, a crucial skill in mathematics.

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Most popular questions from this chapter

Simplify the given expression. $$ 20 \cdot 10+15 \div 5-7 \cdot 6 $$

Simplify the given expression. $$ 9 \cdot(7+7)-8 \cdot(3+8) $$

Simplify the given expression. $$ 6 \cdot 3^{2}+7 \cdot 5+12 $$

$$ 4+9 \cdot[7+6 \cdot(3+3)] $$$$ 15-1[19-(7+3)] $$

Create factor trees for each number. Write the prime factorization for each number in compact form, using exponents. $$270$$
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