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

After analyzing, it's found that \(10\left(9^{6}\right)\) could be divided perfectly by 100 and 540.

Step-by-step solution

Calculate \(10(9^6)\)

Begin by calculating the value of \(10\left(9^{6}\right)\). Use the exponent law that says \(a^{m} * a^{n} = a^{m+n}\) to simplify \(9^{6}\). After this, multiply the resulting value by 10 to find the ultimate value.

Check for divisibility

Now that you have the value of \(10\left(9^{6}\right)\), the next step will be to determine which of the given number(s) can divide the computed value perfectly (i.e., without any remainder). Do this by taking each option one after another and use it to divide \(10\left(9^{6}\right)\). A number that results in a whole quotient and zero remainder will be a suitable divisor. To save time, start by trying to factorize the options. Any number that has a factor also present in the factorization of \(10(9^6)\) could be a potential correct divisor.

Confirming the Answers

After finding all the numbers that could divide \(10\left(9^{6}\right)\) perfectly, list them as the answer(s). Remember, there could be more than one correct answer.

Key concepts

Integer Division

In mathematics, integer division is the process of dividing two integers to obtain another integer as the result without any fractional parts. The key to integer division is finding whether one integer divides another completely, meaning there is no remainder left after the division. This is critical when exploring the division of numbers in problems like the one concerning the expression \(10(9^6)\).

This operation requires us to test numbers to see if they can completely divide the expression \(10(9^6)\) without leaving a remainder. When you perform integer division, you look for whole number quotients, which is important in determining which numbers can produce an integer result when dividing this expression.

Divisibility Rules

Divisibility rules are simple shortcuts that help us quickly determine whether one number divides another without doing long division. These rules can save time and make calculations much easier. For example, for a number to be divisible by 5, it must end in 0 or 5.

In the exercise, divisibility rules are essential for quickly checking whether the options can divide \(10(9^6)\). By understanding the factorization of \(10(9^6)\), you can apply divisibility rules:

• If the number is divisible by 9, the sum of its digits must divide by 9.
• If it ends in zero, it is divisible by 10.
• For divisibility by larger numbers, you need to break them into smaller divisors that comply with known rules.

These rules enable faster verification of possible divisibility.

Exponent Laws

Exponent laws define how to handle mathematical operations involving exponents. One of the most fundamental laws is the multiplication of powers with the same base, given by \(a^m \cdot a^n = a^{m+n}\). This law helps simplify complex expressions involving exponents.

For the expression \(10(9^6)\), applying exponent laws is crucial. Here, the number 9 is expressed as a power of 3, since \(9 = 3^2\). Then, \(9^6\) becomes \((3^2)^6 = 3^{12}\). Applying exponent laws allows us to better understand the number's true value and quickly confirm its divisibility by simplifying it before multiplying by 10.
By mastering exponent laws, we can manipulate and understand numbers more deeply.

Number Factorization

Number factorization involves breaking down a number into a set of base numbers multiplied together to create the original number. This process is helpful in identifying divisors of a number or expression, such as \(10(9^6)\).

Factorization is used in step 2 of the problem-solving process to check potential divisors. By breaking down \(10(9^6)\), we recognize that it is made up of \(10\) and \(3^{12}\) (from simplifying \(9^6\)). These factors reveal possible divisibility by other numbers. For example, from the factorization, we know \(10\) is \(2 \times 5\) and \(3^2\) is a component of \(9^6\). If a number's factorization shares these factors, it might divide the expression with no remainder.

• Understanding the factors of your number makes it easier to identify potential whole divisors.
• Breaking down larger numbers into smaller prime factors can highlight simplified relationships with other numbers.

Mastering factorization is a powerful tool for solving divisibility problems.