Fundamental Math Rules: Powers, Roots, and Divisibility

Rule of Signs

When performing operations, follow these sign rules:

• (+) and (+) = +
• (+) and (-) = -
• (-) and (-) = +
• (-) and (+) = -

Calculation of Powers

To calculate a power, multiply the base by itself as many times as indicated by the exponent.

• 4 raised to 1 = 4
• 45 raised to 1 = 45
• -4² = -(4 · 4) = -16
• 3² = 3 · 3 = 9
• (-4)³ = (-4) · (-4) · (-4) = -64 (Note: - · - = + and + · - = -)

Operations with Powers

When operations involve the same base, the following rules apply:

• Multiplication: The exponents are added.
• Division: The exponents are subtracted.
• Parentheses: The exponents are multiplied.

Square Roots

To find a square root, we look for a number that, when squared, most closely approximates the root. The base of this power is the result. To compute the remainder, subtract the square of the result from the original number.

Operations with Integers

Always remember the rule of signs when adding or subtracting integers. For example: (-7) + (-4) = -7 - 4 = -11.

Combined Operations

Follow the standard order of operations to perform calculations correctly:

1. Parentheses () or brackets []
2. Powers and Roots
3. Multiplication and Division
4. Addition and Subtraction

Divisibility Criteria

• 2: If the last digit is even or 0.
• 3: If the sum of its digits is a multiple of 3.
• 5: If the last digit is 5 or 0.
• 10: If the number ends in 0.
• 11: If the alternating sum of the digits results in 0 or a multiple of 11.
  Example: 121 → 1 + 1 = 2; 2 - 2 = 0 (divisible by 11).

Multiples and Divisors

Multiples

Example: Multiples of 7: M(7) = {7, 14, 21, ...}. This is essentially the multiplication table of the number.

Divisors

Example: Divisors of 21: D(21) = {1, 3, 7, 21}. These are numbers that divide the original number evenly. Use divisibility criteria to identify them.

Prime Numbers and Factorization

Prime Numbers: These are positive numbers whose only divisors are themselves and 1. Examples: 2, 3, 5, 7, 11, 13, 17...

Factorization: This involves breaking a number down into its smallest prime factors. Divide the number starting from the smallest possible prime factor.

Least Common Multiple and Greatest Common Divisor

Least Common Multiple (LCM/MCM)

Example: LCM(12, 16) = 2⁴ · 3 = 16 · 3 = 48

• 12 = 2² · 3
• 16 = 2⁴

To find the LCM, take all unique factors with the largest exponent and multiply them.

Greatest Common Divisor (GCD/MCD)

Example: GCD(28, 56) = 2² · 7 = 28

• 28 = 2² · 7
• 56 = 2³ · 7

To find the GCD, take only the common factors with the smallest exponent and multiply them.

Solving Problems with GCD and LCM

• If the problem requires a result that is a larger number, use LCM.
• If the problem requires a result that is a smaller number, use GCD.