Question

By which of the following could \(10\left(9^{6}\right)\) be divided by to produce an integer result? Indicate all such values. a. 90 b. 100 c. 330 d. 540 e. 720

Short answer

The value \(10(9^{6})\) is divisible by option a, which is 90, and no other options.

Step-by-step solution

Calculate the value of \(10(9^{6})\)

The first step is to calculate the value of \(10(9^{6})\). To do this, raise 9 to the power of 6, and then multiply the result by 10. This gives us \(10(531441) = 5314410\).

Determine divisibility by potential factors

The next step involves testing each of the given options to see if \(5314410\) divided by the option results in an integer. For option a, \(5314410 / 90 = 59049\) which is an integer. For option b, \(5314410 / 100 = 53144.1\) which is not an integer. For option c, \(5314410 / 330 = 16104.2727\) which is not an integer. For option d, \(5314410 / 540 = 9837.80556\) which is not an integer. For option e, \(5314410 / 720 = 7381.125\) which is not an integer.

Key concepts

Divisibility

Divisibility means that one number can be divided by another without leaving a remainder. When thinking about divisibility, imagine splitting a pie equally among friends so that there's no leftover piece. In mathematics, this concept helps us understand how numbers relate to each other.
To check if a number is divisible by another, divide it and see if the quotient is an integer. In our exercise, we checked if 5314410 can be perfectly divided by the options given. If the division gives a whole number, it confirms divisibility.
For example:

• 5314410 divided by 90 equals 59049, a whole number, proving divisibility.
• 5314410 divided by 100 results in 53144.1, which isn't a whole number, indicating non-divisibility.

Using divisibility helps determine if calculations yield neat results or if adjustments are needed.

Integer Results

An integer is a whole number without fractions or decimals. In mathematical problems like our exercise, achieving an integer result is crucial to confirming that one number divides perfectly into another.
Ensuring integer results is often a goal in calculations involving division, particularly in standardized tests like the GRE. When performing a division, if the quotient is a whole number, this means the division was exact, with no remainder left over.
For example, when we divided 5314410 by 90, the result was 59049, clearly an integer. However, dividing it by 100 resulted in 53144.1, which contains a decimal and is therefore not an integer.
Always check for integer results to confirm accurate and tidy calculations.

Exponents

Exponents indicate repeated multiplication of a number by itself. A simple way to think of exponents is as a shorthand for multiple instances of a factor being used together.
In our exercise, we encountered the number 9 raised to the power of 6, represented as \(9^6\). This means multiplying 9 by itself 5 more times: \(9 * 9 * 9 * 9 * 9 * 9\).
Exponents can grow numbers rapidly, helping with complex calculations by condensing repeated multiplications into a single expression. Thus, \(9^6\) equals 531441, a large number derived from relatively simple operations. This result was then multiplied by 10, creating a larger product to test divisibility against the given options.
Understanding exponents can simplify handling big numbers and facilitate accurate results in solving problems.

Multiplication Calculation

Multiplication is a fundamental mathematical operation that combines repeated addition. In many problems, it serves as the backbone technique to compute larger values from smaller ones.
In our scenario, calculating \(10(9^6)\) involved multiplying 531441 (the result of \(9^6\)) by 10, giving 5314410. This result was foundational in testing divisibility by various options.
To perform strong multiplication calculations, remember:

• Ensure all numbers are correctly expanded or simplified before multiplication.
• Double-check each step to prevent carrying forward errors from calculations with large numbers.

Being precise with multiplication ensures that subsequent operations, like division for divisibility, are based on correct values, crucial for reaching valid solutions.