HCF and LCM

\( \begin{aligned} & \text{Important Formula:} \\ & \text{HCF} = \text{Common Factor} \\ & \text{Note:} \\ & \text{HCF will be ‘1’ if no common factor exists.} \end{aligned} \)

\( \begin{aligned} & \text{Q1. Find the HCF of:} (x^2 - 9), (x^3 + 27) \\ & Sol^n: \text{Here, First Expression: } \\ & = (x^2 - 9) \\ & = x^2 – 3^2 \\ & = (x + 3) (x-3) \\ \\ & \text{Also, Second Expression:} \\ & = (x^3 + 27) \\ & = x^3 + 3^3 \\ & = (x + 3) (x^2 + x \cdot 3 + 3^2) \\ & = (x + 3) (x^2 + 3x + 9) \\ & \therefore \text{HCF} = \text{Common Factor} = (x + 3) \end{aligned} \)

\( \begin{aligned} & \text{Q2. Find the HCF of: } (2a^3 - a^2 + a - 2), (a^3 – a^2 + a - 1) \\ & Sol^n: \text{Here, First Expression: } \\ & = (2a^3 - a^2 + a - 2) \\ & = 2a^3 – 2 - a^2 + a \\ & = 2(a^3 – 1) – a(a - 1) \\ & = 2\lbrace (a – 1)(a^2 + a \cdot 1 + 1^2 )\rbrace – a(a - 1) \\ & = 2\lbrace (a – 1)(a^2 + a + 1)\rbrace – a(a - 1) \\ & = (a - 1)\lbrace 2(a^2 + a + 1) – a \rbrace \\ \\ & \text{Second Expression:} \\ & = (a^3 – a^2 + a - 1) \\ & = (a^3 – 1^3 - a^2 + a) \\ & = (a - 1) (a^2 + a \cdot 1 + 1^1) – a(a - 1)) \\ & = (a - 1) (a^2 + a + 1) – a(a - 1)) \\ & = (a - 1) (a^2 + a + 1 – a) \\ & = (a - 1) (a^2 + 1) \\ & \therefore \text{HCF} = \text{Common Factor} = (a - 1) \end{aligned} \)

\( \begin{aligned} & \text{Q3. Find the HCF of: } (a^3 - b^3), (a^3 – a^2b + ab^2), (a^4 + a^2b^2 + b^4) \\ & Sol^n: \text{Here, First Expression: } \\ & = (a^3 - b^3)\\ & = (a + b)(a^2 - ab + b^2) \\ \\ & \text{Also, Second Expression:} \\ & = (a^3 – a^2b + ab^2) \\ & = a (a^2 – ab + b^2) \\ \\ & \text{And, Third Expression:} \\ & = (a^4 + a^2b^2 + b^4) \\ & = (a^2 + ab + b^2) (a^2 – ab + b^2) \\ & \therefore \text{HCF} = \text{Common Factor} = (a^2 – ab + b^2) \end{aligned} \)

\( \begin{aligned} & \text{Important Formula:} \\ & \text{LCM} = \text{Common Factor } \times \text{ Remaining Factor} \end{aligned} \)

\( \begin{aligned} & \text{Q1. Find the LCM of: } (a^3 – y^3), (a^4 + a^2y^2 + y^4) \\ & Sol^n: \text{Here, First Expression: } \\ & = (a^3 – y^3) \\ & = (a - y)(a^2 + ay + y^2) \\ \\ & \text{Also, Second Expression:} \\ & = (a^4 + a^2y^2 + y^4)  \\ & = (a^2 + ay + y^2) (a^2 - ay + y^2) \\ & \text{Now, Common Factor} = (a^2 + ay + y^2) \\ & \text{And, Remaining Factor} = (a - y)(a^2 - ay + y^2) \\ & \text{So,} \text{LCM} = \text{Common Factor} \times \text{Remaining Factor} \\ & \therefore \text{LCM} = (a^2 + ay + y^2) (a - y)(a^2 - ay + b^2) \end{aligned} \)

\( \begin{aligned} & \text{Q2. Find the LCM of: } (\frac{x^4}{y^4} + \frac{y^4}{x^4} + 1), (\frac{x^3}{y^3} + \frac{y^3}{x^3})\\ & Sol^n: \text{Here, First Expression: } \\ & = (\frac{x^4}{y^4} + \frac{y^4}{x^4} + 1) \\ & = (\frac{x^2}{y^2} + \frac{y^2}{x^2} + 1) (\frac{x^2}{y^2} + \frac{y^2}{x^2} - 1) \\ \\ & \text{Also, Second Expression:} \\ & = (\frac{x^3}{y^3} + \frac{y^3}{x^3}) \\ & = (\frac{x}{y} + \frac{y}{x})(\frac{x^2}{y^2} + \frac{y^2}{x^2} + 1) \\ & \text{Now, Common Factor} = (\frac{x^2}{y^2} + \frac{y^2}{x^2} + 1) \\ & \text{And, Remaining Factor} = (\frac{x}{y} + \frac{y}{x})(\frac{x^2}{y^2} + \frac{y^2}{x^2} - 1) \\ & \text{So,} \text{LCM} = \text{Common Factor} \times \text{Remaining Factor} \\ & \therefore \text{LCM} = (\frac{x^2}{y^2} + \frac{y^2}{x^2} + 1) (\frac{x}{y} + \frac{y}{x})(\frac{x^2}{y^2} + \frac{y^2}{x^2} - 1) \end{aligned} \)

\( \begin{aligned} & \text{Q3. Find the LCM of: } (x^3 + 1), (x^4 – x^3 + x^2), (x^3 – x^2 + x ) \\ & Sol^n: \text{Here, First Expression: } \\ & = (x^3 + 1) \\ & = (x + 1)(x^2 + x + 1) \\ \\ & \text{Also, Second Expression:} \\ & = (x^4 – x^3 + x^2) \\ & = x^2(x^2 – x + 1) \\ \\ & \text{And, Third Expression:} \\ & = (x^3 – x^2 + x )\\ & = x(x^2 – x + 1) \\ \\ & \text{Now, Common Factor} = 1 \\ & \text{And, Remaining Factor} = x^2(x + 1)(x^2 + x + 1) (x^2 – x + 1)  \\ & \text{So,} \text{LCM} = \text{Common Factor} \times \text{Remaining Factor} \\ & \therefore \text{LCM} = x^2(x + 1)(x^2 + x + 1) (x^2 – x + 1) \end{aligned} \)