3.3 Number Theory, Exponents, and Roots

Foundations of Number Theory

Number theory focuses on the properties of integers. On the SHSAT, number theory concepts are often combined with algebra to test mathematical reasoning. Concepts such as factors, multiples, prime numbers, greatest common factor (GCF), and least common multiple (LCM) form the basis of these questions.

Prime and Composite Numbers

• A prime number is an integer greater than 1 that has exactly two positive factors: 1 and itself. The first ten prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Note that 2 is the only even prime number, and 1 is neither prime nor composite.
• A composite number is an integer greater than 1 that has more than two factors.

Prime Factorization

The Fundamental Theorem of Arithmetic states that every composite number can be uniquely factored into a product of prime numbers. Prime factorization is the process of breaking down a composite number into its prime building blocks. Example: Find the prime factorization of 120. Using a factor tree: 120 = 12 * 10 = (3 * 4) * (2 * 5) = 3 * (2 * 2) * 2 * 5 = 2^3 * 3 * 5.

Greatest Common Factor (GCF) and Least Common Multiple (LCM)

Greatest Common Factor (GCF)

The GCF of two or more integers is the largest integer that divides each of them without a remainder. Method using Prime Factorization: Write the prime factorization of each number. The GCF is the product of the lowest power of each common prime factor. Example: Find the GCF of 72 and 90.

• 72 = 2^3 * 3^2
• 90 = 2 * 3^2 * 5
• Common prime factors are 2 and 3. The lowest power of 2 is 2^1. The lowest power of 3 is 3^2.
• GCF = 2^1 * 3^2 = 2 * 9 = 18.

Least Common Multiple (LCM)

The LCM of two or more integers is the smallest positive integer that is a multiple of each of them. Method using Prime Factorization: Write the prime factorization of each number. The LCM is the product of the highest power of all prime factors present in either number. Example: Find the LCM of 72 and 90.

• 72 = 2^3 * 3^2
• 90 = 2 * 3^2 * 5
• Prime factors present are 2, 3, and 5. The highest power of 2 is 2^3. The highest power of 3 is 3^2. The highest power of 5 is 5^1.
• LCM = 2^3 * 3^2 * 5 = 8 * 9 * 5 = 360.

Real-World Applications

• GCF Problems: Look for key phrases like "divide into equal groups," "cut into equal pieces," or "maximum length."
• LCM Problems: Look for scenarios where events repeat at different intervals and you need to find when they will occur at the same time again (e.g., traffic lights flashing, buses departing, alarms ringing).

Rules of Exponents

Exponents are shorthand notation for repeated multiplication. You must memorize and apply the following exponent rules:

1. Product Rule: x^a * x^b = x^(a+b)
2. Quotient Rule: x^a / x^b = x^(a-b)
3. Power of a Power Rule: (x^a)^b = x^(ab)
4. Power of a Product Rule: (xy)^a = x^a * y^a
5. Negative Exponent Rule: x^-a = 1 / x^a
6. Zero Exponent Rule: x^0 = 1 (for x != 0)

Example: Simplify (2^5 * 3^4) / (2^3 * 3^5). (2^5 * 3^4) / (2^3 * 3^5) = 2^(5-3) * 3^(4-5) = 2^2 * 3^-1 = 4 * (1/3) = 4/3.

Roots: Square Roots and Cube Roots

A square root of a number is a value that, when multiplied by itself, gives the original number. The symbol sqrt(x) represents the principal (positive) square root of x. A cube root, denoted by cubert(x), is a value that, when multiplied by itself three times, gives the original number.

Perfect Squares and Cubes

Students should memorize perfect squares up to 15^2 = 225 and perfect cubes up to 5^3 = 125:

• Perfect Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
• Perfect Cubes: 1, 8, 27, 64, 125

Simplifying Radicals

To simplify a square root, factor out the largest perfect square divisor: sqrt(48) = sqrt(16 * 3) = sqrt(16) * sqrt(3) = 4 * sqrt(3).

Estimating Non-Perfect Squares

If a radical is not a perfect square, estimate its value by finding the two consecutive perfect squares it lies between. Example: Estimate sqrt(75). Since 64 < 75 < 81, it follows that: sqrt(64) < sqrt(75) < sqrt(81), which means 8 < sqrt(75) < 9. Because 75 is closer to 81 than to 64, sqrt(75) is approximately 8.7.

Parity and Divisibility Rules

Parity refers to whether an integer is even or odd. Keep these algebraic relationships in mind:

• Even +/- Even = Even
• Odd +/- Odd = Even
• Even +/- Odd = Odd
• Even * Even = Even
• Odd * Odd = Odd
• Even * Odd = Even

Divisibility rules help quickly determine factors without long division:

• Divisible by 3: The sum of the digits is divisible by 3 (e.g., for 273, 2+7+3 = 12, which is divisible by 3).
• Divisible by 4: The last two digits form a number divisible by 4 (e.g., 516 is divisible by 4 because 16 is).
• Divisible by 6: The number is divisible by both 2 and 3.
• Divisible by 9: The sum of the digits is divisible by 9.

Scientific Notation

Scientific notation expresses very large or very small numbers as the product of a decimal between 1 and 10 (inclusive of 1, exclusive of 10) and a power of 10. The form is:

$a \times 10^n, \quad 1 \le |a| < 10, \quad n \text{ an integer}$

Converting to Scientific Notation

Example: Write 45,000 in scientific notation. Move the decimal left until one non-zero digit remains to its left: 4.5. The decimal moved 4 places, so n = 4. The result is 4.5 × 10^4.

Example: Write 0.00067 in scientific notation. Move the decimal right until one non-zero digit remains to its left: 6.7. The decimal moved 4 places to the right, so n = -4. The result is 6.7 × 10^(-4).

Multiplying and Dividing in Scientific Notation

Use exponent rules to combine the powers of 10, then adjust the decimal so the coefficient stays between 1 and 10.

• (a × 10^m)(b × 10^n) = (ab) × 10^(m+n)
• (a × 10^m) / (b × 10^n) = (a/b) × 10^(m-n)

Example: (3 × 10^4)(2 × 10^5) = 6 × 10^9. If instead (4 × 10^4)(5 × 10^3) = 20 × 10^7, rewrite 20 × 10^7 as 2.0 × 10^8 to keep the coefficient between 1 and 10.

Adding and Subtracting in Scientific Notation

First rewrite both numbers so they share the same power of 10, then add or subtract the coefficients. Example: 3.2 × 10^5 + 4.1 × 10^4. Rewrite 4.1 × 10^4 as 0.41 × 10^5. Then 3.2 × 10^5 + 0.41 × 10^5 = 3.61 × 10^5.

SHSAT Applications

Scientific notation appears on the SHSAT in astronomy, biology, and measurement problems involving very large or very small quantities. Common traps include forgetting to adjust the coefficient after multiplication (leaving a coefficient of 20 instead of 2.0 × 10^1) and miscounting the number of decimal-place moves when converting.