1. Find the H.C.F. and L.C.M. of $3x^6y$ and $15x^5y^3$. 2. Find the H.C.F. and L.C.M. of $25pq^2r^3$ and $10p^3qr^2$. 3. Find the H.C.F. and L.C.M. of $x^3y$, $x^2y^5$, and $x^4y^4$. 4. Find the H.C.F. and L.C.M. of $a^4b$, $b^3c^2$, and $c^6a^2$. --- ### Step 1: Understanding H.C.F. and L.C.M. of polynomials - The **H.C.F.** (Highest Common Factor) is the polynomial of the highest degree that divides all given polynomials. - The **L.C.M.** (Least Common Multiple) is the polynomial of the lowest degree that all given polynomials divide into. **Rule:** For each variable, choose the lowest power for H.C.F. and the highest power for L.C.M. --- ### Problem 1: $3x^6y$ and $15x^5y^3$ 1. Factor each term: $$3x^6y = 3 \times x^6 \times y$$ $$15x^5y^3 = 15 \times x^5 \times y^3$$ 2. Find H.C.F.: - Coefficients: H.C.F. of 3 and 15 is 3 - $x$: minimum power is $x^5$ - $y$: minimum power is $y^1$ So, $$\text{H.C.F.} = 3x^5y$$ 3. Find L.C.M.: - Coefficients: L.C.M. of 3 and 15 is 15 - $x$: maximum power is $x^6$ - $y$: maximum power is $y^3$ So, $$\text{L.C.M.} = 15x^6y^3$$ --- ### Problem 2: $25pq^2r^3$ and $10p^3qr^2$ 1. Factor each term: $$25pq^2r^3 = 25 \times p \times q^2 \times r^3$$ $$10p^3qr^2 = 10 \times p^3 \times q \times r^2$$ 2. Find H.C.F.: - Coefficients: H.C.F. of 25 and 10 is 5 - $p$: minimum power is $p^1$ - $q$: minimum power is $q^1$ - $r$: minimum power is $r^2$ So, $$\text{H.C.F.} = 5pq r^2$$ 3. Find L.C.M.: - Coefficients: L.C.M. of 25 and 10 is 50 - $p$: maximum power is $p^3$ - $q$: maximum power is $q^2$ - $r$: maximum power is $r^3$ So, $$\text{L.C.M.} = 50p^3q^2r^3$$ --- ### Problem 3: $x^3y$, $x^2y^5$, $x^4y^4$ 1. Factor each term: $$x^3y = x^3 \times y^1$$ $$x^2y^5 = x^2 \times y^5$$ $$x^4y^4 = x^4 \times y^4$$ 2. Find H.C.F.: - $x$: minimum power is $x^2$ - $y$: minimum power is $y^1$ So, $$\text{H.C.F.} = x^2 y$$ 3. Find L.C.M.: - $x$: maximum power is $x^4$ - $y$: maximum power is $y^5$ So, $$\text{L.C.M.} = x^4 y^5$$ --- ### Problem 4: $a^4b$, $b^3c^2$, $c^6a^2$ 1. Factor each term: $$a^4b = a^4 \times b^1$$ $$b^3c^2 = b^3 \times c^2$$ $$c^6a^2 = c^6 \times a^2$$ 2. Find H.C.F.: - $a$: powers are 4, 0, 2; minimum is 0 (since $b^3c^2$ has no $a$) - $b$: powers are 1, 3, 0; minimum is 0 (since $c^6a^2$ has no $b$) - $c$: powers are 0, 2, 6; minimum is 0 (since $a^4b$ has no $c$) So, $$\text{H.C.F.} = 1$$ (no common variable factor) 3. Find L.C.M.: - $a$: maximum power is $a^4$ - $b$: maximum power is $b^3$ - $c$: maximum power is $c^6$ So, $$\text{L.C.M.} = a^4 b^3 c^6$$