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Chapter 1: Problem 8

64 is not a multiple of 6

Short Answer

Expert verified
64 is not a multiple of 6 because there is a remainder when divided.

Step by step solution

01

Determine Multiples of 6

To determine if 64 is a multiple of 6, we first need to recall what multiples of 6 are. A multiple of 6 is any number that can be expressed as \(6 \times n\), where \(n\) is an integer. Some of the multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, etc.
02

Check the Division

For a number to be a multiple of another, when you divide it by the latter, the remainder should be zero. Let's check the division of 64 by 6. When we divide 64 by 6, the quotient is 10 and the remainder is 4.
03

Evaluate Remainder

Since the remainder when dividing 64 by 6 is not zero (it is 4 in this case), 64 is not evenly divisible by 6. This confirms that 64 is not a multiple of 6.

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Key Concepts

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

Understanding Division
Division is a fundamental concept in mathematics that allows us to see how many times one number is contained within another. This process involves dividing a dividend by a divisor to obtain a quotient. For example, when determining if number 64 is a multiple of 6, we divide it by 6.

The division can be expressed as follows: if you take 64 and divide it by 6, the result will be a quotient of 10 and a remainder of 4. This tells us that 6 goes into 64 a total of 10 times with 4 left over.
  • The dividend is 64.
  • The divisor is 6.
  • The quotient is 10.
  • The remainder is 4.
The key takeaway is that for a number to be a complete multiple of another, the division should yield no remainder.
Exploring Integer Multiplication
Multiplication is the process where we add a number to itself a certain number of times. When talking about integer multiplication in the context of multiples, it involves multiplying an integer by a whole number.

Say that number is 6. If you multiply 6 by different integers, the products are known as multiples of 6. For instance, multiplying 6 by 1 gives 6, by 2 gives 12, by 3 gives 18, and so forth.
  • 6 times 1 equals 6
  • 6 times 2 equals 12
  • 6 times 3 equals 18
  • ... and so on
So in the context of our example, 64 should ideally be reachable by multiplying 6 by some integer without any leftover, though in this case, since a remainder exists, 64 isn't exactly a multiple of 6.
Understanding Remainder in Division
The remainder is what is left after you divide one number by another and it doesn't go in evenly. It signifies that the division process has completed, but not perfectly.

In our example, when trying to see if 64 is a multiple of 6, dividing 64 by 6 leaves a remainder of 4. This leftover indicates that 64 does not perfectly split into groups of 6.
  • A remainder of 0 implies the number is a perfect multiple
  • A non-zero remainder, like 4, shows the number is not a perfect multiple
In summary, the remainder is a critical component in determining whether one number is a multiple of another, as it tells us about the evenness of the division process.

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Most popular questions from this chapter

For Problems 63-74, find the greatest common factor of the given numbers. $$ 84 \text { and } 96 $$

Simplify each numerical expression. Don't forget to take advantage of the properties if they can be used to simplify the computation. $$-24+18+19-30$$

For Problems 51-62, factor each composite number into a product of prime numbers. For example, \(18=2 \cdot 3 \cdot 3\). 135

$$ \begin{array}{r} 17 \\ -19 \end{array} $$

Simplify each algebraic expression by combining similar terms. $$-a b-a+4 a b+7 a b-3 a-11 a b$$
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