Relationship between LCM and GCD

Q. Find the L.C.M. of the two expressions $a^2+7a-18$ and $a^2+10a+9$ with the help of their H.C.F.

1. $(a-2)(a+9)(a+1)$
2. $(a-2{)}^2(a+9\left)\right(a+1)$
3. $(a-2)(a+9{)}^2(a+1)$
4. $(a-2)(a+9)(a-9)$

Q. Question 40
Give a possible expression for the length and breadth of the rectangle whose area is given by $4a^2+4a-3$.

Q.

The product of the GCD and LCM of any two polynomials is equal to the __ of those two polynomials.

Q. $m^2-5m-14$ is an expression. Find out another similar expression such that their H.C.F. is $(m-7)$ and L.C.M. is $m^3-10m^2+11m+70$.

1. $m^2-12m-35$
2. $m^2+12m+35$
3. $m^2-12m+35$
4. $m^2+12m-35$

Q. Question 2
Prove that one and only one out of n, (n + 2) and (n + 4) is divisible by 3, where n is any positive integer.

Q. Question 3
5x + 9 = 5 + 3x

Q. If the LCM of two polynomials is $(x-1{)}^2(x-2{)}^2$ and their HCF is $(x-1)(x-2)$, then the product of the polynomials is:

1. $(x-1{)}^3(x-2{)}^3$
2. $(x-1{)}^3(x-2{)}^2$
3. $(x-1{)}^2(x-2{)}^3$
4. $(x-1)(x-2)$

Q.

Let GCD $=x+1$ and LCM $=x^6-1$ for two given polynomials $f\left(x\right)$ and $g\left(x\right)$. Then, if $f\left(x\right)=x^3+1$. what is $g\left(x\right)$?

1. $(x^3+1)(x-1)$

2. $(x^3-1)(x+1)$

3. $x$

4. $x^2+1+x^3$

Q. Let the GCD =x+1 and LCM =$x^6-1$ for gives two polynomials. Then, if $f\left(x\right)=x^3+1$. what is g(x)?

1. x
   
2. $(x^3-1)(x+1)$
   
3. $(x^3+1)(x-1)$
   
4. $x^2+1+x^3$

Q. Product of the GCD and LCM of polynomials = ______

1. subtraction of polynomials
2. product of polynomials
3. None of the above
4. sum of polynomials

Q.

Using Euclids algorithm, find the HCF of $405$ and $2520$

Q. Find the LCM of the following polynomials
$a^2-3a+2\mathrm{and}{a}^2+2a-8$ whose GCD is $(a-2)$

1. $\left(a+1\right)\left(a+2\right)\left(a-4\right)$
2. $\left(a-1\right)\left(a+2\right)\left(a+4\right)$
3. $\left(a-1\right)\left(a-2\right)\left(a-4\right)$
4. $\left(a-1\right)\left(a-2\right)\left(a+4\right)$

Q. Find the other polynomial q (x) of each of the following, given that LCM and GCD and one polynomial p(x) respectively. $2(x+1)(x^2-4),(x+1),(x+1)(x-2)$

1. $2(x+1)(x+2)$
2. $2(x-1)(x+2)$
3. $2(x+1)(x-2)$
4. None of these

Q.

A sweet seller has $420$ Rasgullahs and $130$ Laddus. He wants to stack them in such a way that each stack has the same number, and they take up the least area of the tray. What is the number of sweets that can be placed in each stack for this purpose?

Q. The GCD of two polynomials is $x{y}^2{z}^2$ and their LCM is $x{y}^3{z}^8$. If one of the polynomials is $x{y}^2{z}^2$, then the other one is .

1. $x{y}^3{z}^8$
2. $x{y}^2{z}^6$
3. $x{y}^3{z}^6$
4. $x{y}^2{z}^8$

Q.

Find the least common multiple of the following polynomials $9\left(x+2\right)\left(2x-1\right)$ and $3\left(x+2\right)$.

Q. If $u\left(x\right)=(x-1{)}^2$ and $v\left(x\right)=(x^2,-1)$ then verify $LCM\times HCF=u\left(x\right)\times v\left(x\right)$.

Q. For two polynomial p(x) and q(x), their H.C.F. h(x) $=$ x-2 and their L.C.M. $m\left(x\right)=x^2-3x+2$, find the other polynomial q(x).

1. $x^2-x-6$
2. $x^2+x+6$
3. $x^2+x-6$
4. $x^2-x+6$

Q. Find the LCM of the following: $x^2-5x+6,x^2+4x-12$ whose GCD is $x-2$

1. $(x-3)(x-2)(x+6)$
2. $(x+3)(x-2)(x+6)$
3. $(x-3)(x+2)(x+6)$
4. None of these

Q. If the LCM of two polynomials is $(x-1{)}^2(x-2{)}^2$ and their HCF is $(x-1)(x-2)$, then the product of the polynomials is:

1. $(x-1{)}^3(x-2{)}^3$
2. $(x-1{)}^3(x-2{)}^2$
3. $(x-1{)}^2(x-2{)}^3$
4. $(x-1)(x-2)$

Q. The H.C.F. of two expressions is x and their L.C.M is $x^3-9x$ IF one of the expression is $x^2+3x$ then, the other expression is

1. $x^2-3x$
2. $x^3-3x$
3. $x^2+9x$
4. $x^2-9x$

Q. Find the HCF $h\left(x\right)$ of the following polynomials and using HCF $h\left(x\right)$, find the LCM $m\left(x\right)$, the polynomials are:$3x^2+5x-2,3x^2-7x+2$

Q. If $A\times B=H\times L$, then $L=$

1. $\frac{A\times H}{B}$
2. $\frac{A\times B}{H}$
3. $\frac{H}{A\times B}$
4. $\frac{B\times H}{A}$

Q.

Find the value of $(-2a^2)\times (-3b^2)\times (-4c^2)\times abc$

Q. Find the LCM of the following: $x^3{y}^2,xyz$

1. $x^3{y}^{2}z$
2. $x^2{y}^{2}z$
3. $x^3{y}^2{z}^2$
4. None of these

Q. Find the LCM of the following: $2x^3+15x^2+2x-35,x^3+8x^2+4x-21$ whose GCD is $x+7$.

1. $(2x^2+x-5)(x^3+8x^2+4x-21)$
2. $(2x^2-x-5)(x^3+8x^2+4x-21)$
3. $(2x^2+x-5)(x^3-8x^2+4x-21)$
4. None of these

Q. The HCF of two polynomials is $h\left(a\right)=a-7$ and their LCM is $m\left(a\right)=a^3-10a^2+11a+70$. If one of the polynomial is $p\left(a\right)=a^2-12a+35$, and if the leading coefficient of $q\left(a\right)$ is positive, find the other polynomial $q\left(a\right)$.

Q. The product of two expressions is $a^4-9a^2+4a+12$ and their H.C.F. is $a-2$. Find their L.C.M.

Q. Find the other polynomial q (x) of the following, given that LCM, GCD and one polynomial p(x) respectively. $(4x+5{)}^3(3x-7{)}^3,(4x+5)(3x-7{)}^2,(4x+5{)}^3(3x-7{)}^2$

1. $(3x-7{)}^3(4x+5)$
2. $(3x+7{)}^3(4x+5)$
3. $(3x-7{)}^3(4x-5)$
4. None of these

Q. Find the other polynomial q (x) of the following, given that LCM, GCD and one polynomial p(x) respectively. $(x^3-4x)(5x+1),(5x^2+x),(5x^3-9x^2-2x)$.

1. $x(x+2)(5x+1)$
2. $x(x+2)(5x-1)$
3. $x(x-2)(5x+1)$
4. None of these