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Chapter 2: Problem 20

Is 3,269,283 a multiple of \(3 ?\) How do you know?

Short Answer

Expert verified
Yes, because the sum of the digits (33) is a multiple of 3.

Step by step solution

01

Understand the Rule for Multiples of 3

A number is a multiple of 3 if the sum of its digits is also a multiple of 3. Therefore, we need to sum the digits of the number 3,269,283.
02

Sum the Digits

Add up the digits of the number 3,269,283: \[ 3 + 2 + 6 + 9 + 2 + 8 + 3 \ 3 + 2 = 5 \ 5 + 6 = 11 \ 11 + 9 = 20 \ 20 + 2 = 22 \ 22 + 8 = 30 \ 30 + 3 = 33 \] The sum of the digits is 33.
03

Determine if the Sum is a Multiple of 3

Next, check if the sum, 33, is a multiple of 3. Since 33 divided by 3 equals 11, which is an integer, 33 is a multiple of 3.
04

Conclude if the Original Number is a Multiple of 3

Since the sum of the digits of 3,269,283 is a multiple of 3, the original number is also a multiple of 3.

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Key Concepts

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

Sum of Digits
To determine if a number is a multiple of 3, we'll use the sum of its digits. For our exercise, the number is 3,269,283. First, we need to break it down digit by digit: 3, 2, 6, 9, 2, 8, 3. Adding these digits together: 3 + 2 = 5, 5 + 6 = 11, 11 + 9 = 20, 20 + 2 = 22, 22 + 8 = 30, and finally, 30 + 3 = 33. The sum of the digits for the number 3,269,283 is 33. Each step in this calculation is crucial to ensure we add correctly. Taking one step at a time avoids mistakes and ensures accuracy.
Divisibility Rules
Divisibility rules help us quickly determine if numbers are divisible by other numbers without performing full division. The rule for 3 is straightforward: a number is divisible by 3 if the sum of its digits is also divisible by 3. For example, for the number 3,269,283, we saw that the sum of its digits is 33. Checking if 33 is divisible by 3, we divide: 33 梅 3 = 11. Since 11 is an integer, 33 is indeed divisible by 3. Therefore, 3,269,283 is a multiple of 3. Learning these rules can simplify many math problems and is a valuable tool for mental math.
Step-by-Step Solution
Let's break down the solution step-by-step to understand why 3,269,283 is a multiple of 3:
1. Understand the rule: A number is a multiple of 3 if the sum of its digits is a multiple of 3.
2. Sum the digits: The digits of 3,269,283 added together (3 + 2 + 6 + 9 + 2 + 8 + 3) equal 33.
3. Check the sum: 33 is divisible by 3 since 33 梅 3 = 11.
4. Conclude: Since the summed number (33) is a multiple of 3, the original number (3,269,283) is also a multiple of 3.
By carefully following each step, we ensured our calculation was correct and easily demonstrated why the original number is a multiple of 3.

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