LCM, HCF, Prime Factors and Index Notation

• LCM stands for Least Common Multiple
• LCM represents the smallest positive integer divisible by all specified numbers
• Listing multiples of each number until a common one is found determines the LCM
• Prime factorization offers another method for LCM calculation
• HCF means Highest Common Factor
• HCF denotes the largest positive integer dividing each of the given integers
• Listing factors of each number to find the largest common one calculates the HCF
• Prime factorization can also determine the HCF

Prime Factors

• Prime factors include the prime numbers dividing a given number exactly
• Every integer above 1 is uniquely expressible as a product of prime numbers
• Finding prime factors involves dividing the number by the smallest prime number that divides it exactly then repeating the process for the quotient
• Factor trees are useful in finding prime factors
• The prime factors of 12 are 2, 2, and 3 as an example

Index Notation Laws

• Index notation provides a way to write repeated multiplication of the same number
• A number in index notation is a base raised to a power, also known as an index
• The expression 2 x 2 x 2 converts to 2^3 in index notation

Multiplication Law

• When multiplying numbers sharing the same base, add the indices
• This is represented as a^m * a^n = a^(m+n)
• For example, 2^2 * 2^3 = 2^(2+3) = 2^5 = 32

Division Law

• When dividing numbers with identical bases, subtract the indices
• The formula is a^m / a^n = a^(m-n)
• For example, 3^5 / 3^2 = 3^(5-2) = 3^3 = 27

Power Law

• When raising a power to another power, multiply the indices
• The formula is (a^m)^n = a^(m*n)
• For example, (2^2)^3 = 2^(2*3) = 2^6 = 64

Zero Index

• Any non-zero number to the power of 0 equals 1
• Expressed as a^0 = 1 (where a ≠ 0)
• For example, 5^0 = 1

Negative Index

• A number to a negative power equals the reciprocal of the number raised to the positive power
• This is shown as a^(-n) = 1 / a^n
• As an example, 2^(-2) = 1 / 2^2 = 1 / 4

Fractional Index

• A number to a fractional power signifies a root
• a^(1/n) equals the nth root of a
• For example, 4^(1/2) equals the square root of 4, which is 2
• a^(m/n) equals the nth root of (a^m), which also equals (nth root of a)^m
• As an example, 8^(2/3) equals the cube root of (8^2), which equals (cube root of 8)^2, which is 2^2 = 4

HCF and LCM using Prime Factorization

• Express each number as a product of its prime factors to find the HCF and LCM
• Multiply the lowest powers of the common prime factors to find the HCF
• Multiply the highest powers of all prime factors present in the numbers to find the LCM
• Example: Finding the HCF and LCM of 12 and 18 involves:
• 12 = 2^2 * 3
• 18 = 2 * 3^2
• HCF = 2^1 * 3^1 = 6
• LCM = 2^2 * 3^2 = 36