1.2 The Fundamental Theorem of Arithmetic

Step 1

Perform prime factorization for each number.

Step 2

$6 = 2^1 \times 3^1$ [cite: 126].

Step 3

$72 = 2^3 \times 3^2$ [cite: 126].

Step 4

$120 = 2^3 \times 3^1 \times 5^1$ [cite: 126].

Step 5

HCF: Identify common factors (2 and 3) and take the smallest power.

Step 6

Smallest power of 2 is $2^1$ . Smallest power of 3 is $3^1$ .

Step 7

HCF = $2^1 \times 3^1 = 6$ [cite: 131].

Step 8

LCM: Identify all prime factors (2, 3, 5) and take the greatest power.

Step 9

Greatest power of 2 is $2^3$ . Greatest power of 3 is $3^2$ . Greatest power of 5 is $5^1$ .

Step 10

LCM = $2^3 \times 3^2 \times 5^1 = 8 \times 9 \times 5 = 360$ [cite: 134].

Final Answer

HCF = 6, LCM = 360

Operation

Number

Quick Reference:

N = Natural Numbers: 1, 2, 3, ...

W = Whole Numbers: 0, 1, 2, ...

Z = Integers: ..., -2, -1, 0, 1, 2, ...

Q = Rational Numbers: p/q form (p, q ∈ Z, q ≠ 0)

R = Real Numbers: Q ∪ Irrational

Select an operation and enter values to see detailed step-by-step solutions.