Question

What is the greatest positive integer $x$ such that $3^x$ is a factor of $9^{10}?$

Short answer

20

Step-by-step solution

Express 9 in terms of base 3

Start by expressing 9 as a power of 3: 9 = 3^2

Raise the expression to the power of 10

Since we need to find the factor of 9 raised to the power of 10, we express it as: (3^2)^{10}

Simplify the exponent

Using the property of exponents (a^{m})^{n} = a^{mn}, we can simplify (3^{2})^{10} = 3^{20}.

Identify the factor

Given that we have 3^{20}, the greatest integer x for which 3^x is a factor of 3^{20} is 20.

Key concepts

Properties of Exponents

When working with exponents, it's essential to understand their core properties. Exponents are a shorthand way to represent repeated multiplication. For instance, expressing 9 as an exponent: 9 can be written as $3^2$. Here are some essential properties of exponents:

**Product of Powers Property**: $a^m\cdot a^n=a^{m+n}$ Combining exponents with the same base by adding them.
**Power of a Power Property**: $(a^m{)}^n=a^{mn}$ Multiplying the exponents when raising a power to another power.

**Quotient of Powers Property**: $a^m/a^n=a^{m-n}$ Subtracting the exponents when dividing with the same base.
Using these properties can simplify many problems, as seen in the exercise solution.

Factorization

Factorization is breaking down numbers or expressions into their simpler parts called factors. For instance, in the problem, we factored 9 into $3^2$. This process helps simplify the calculations and properties.

**Basic Steps in Factorization**:
1. Identify if the number can be expressed as a product of smaller numbers.
2. Break it down sequentially.
3. Look for common factors.
In our exercise, we started by factorizing 9 as $3^2$, enabling us to further simplify the power of 9.

Base Conversion

Base conversion is another essential concept that involves changing the form of numbers.
In our exercise, we expressed 9 in terms of base 3 for easier manipulation: $9=3^2$.

**Steps for Base Conversion**:
1. Identify the conversion needed.
2. Convert by expressing the number using the desired base.
3. Apply suitable exponent properties.
Converting bases help simplify and solve problems, as shown in converting 9 to $3^2$ to find the greater factor of $3^{20}$.