Problem 80 If a number is divisible by 2 an... [FREE SOLUTION]

91影视

Chapter 1: Problem 80

If a number is divisible by 2 and by 3 why is it also divisible by 6?

Short Answer

Expert verified

A number divisible by both 2 and 3 is also divisible by 6.

Step by step solution

Understand Divisibility Rules

A number is divisible by 2 if it is even, and it is divisible by 3 if the sum of its digits is divisible by 3.

Define Given Conditions

The number must be divisible by both 2 and 3. This means it is even and the sum of its digits is divisible by 3.

Analyze Multiples of 2 and 3

The least common multiple (LCM) of 2 and 3 is 6. Therefore, any number divisible by both 2 and 3 must also be divisible by 6.

Conclude Divisibility by 6

Since the number meets the divisibility requirements of both 2 and 3, it must be a multiple of 6.

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

least common multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. To find the LCM of 2 and 3, list the multiples of each number:

Multiples of 2: 2, 4, 6, 8, 10, 12...
Multiples of 3: 3, 6, 9, 12, 15...

The smallest common multiple in these lists is 6. Thus, the LCM of 2 and 3 is 6.

Therefore, a number that is a multiple of both 2 and 3 is also a multiple of 6. This concept is vital in understanding why a number divisible by 2 and 3 must also be divisible by 6.

divisibility by 2

Divisibility rules help us determine if a number can be divided by another number without actually performing the division.

For divisibility by 2, the rule is simple: any even number is divisible by 2. In other words, a number is divisible by 2 if it ends in 0, 2, 4, 6, or 8.

For example:

24 ends in 4, so it is divisible by 2.
37 ends in 7, so it is not divisible by 2.
100 ends in 0, so it is divisible by 2.

Knowing whether a number is even or odd helps you quickly determine if it is divisible by 2.

divisibility by 3

Determining if a number is divisible by 3 involves a straightforward trick: add all the digits of the number and check if the sum is divisible by 3.

For example:

For 123, add the digits: 1 + 2 + 3 = 6. Since 6 is divisible by 3, 123 is also divisible by 3.
For 456, add the digits: 4 + 5 + 6 = 15. Since 15 is divisible by 3, 456 is divisible by 3.
For 789, add the digits: 7 + 8 + 9 = 24. Since 24 is divisible by 3, 789 is divisible by 3.
For 101, add the digits: 1 + 0 + 1 = 2. Since 2 is not divisible by 3, 101 is not divisible by 3.

This method is quick and can easily be applied to any number to check for divisibility by 3.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

If a number is divisible by 2 and by 3 why is it also divisible by 6?

Short Answer

Step by step solution

Understand Divisibility Rules

Define Given Conditions

Analyze Multiples of 2 and 3

Conclude Divisibility by 6

Key Concepts

least common multiple

divisibility by 2

divisibility by 3

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Applied Mathematics

Decision Maths

Mechanics Maths

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.