LCM & HCF — CDS Elementary Mathematics PYQs

What is the HCF of the polynomials $x^8+x^4+1$ and $x^4+x^2+1$?

Consider the following statements in respect of prime numbers $p$ and $q$:
I. Their LCM is always an odd number.
II. Sum of their LCM and HCF is always an even number.
Which of the statements given above is/are correct?

The HCF of $x$ and $y$ is $H$. Consider the following statements in respect of the HCF of $p=\frac{x^3+y^3}{x^2-xy+y^2}$ and $q=\frac{x^3-y^3}{x^2+xy+y^2}$:
I. The HCF of $p$ and $q$ can be $H$.
II. The HCF of $p$ and $q$ can be $2H$.
Which of the statements given above is/are correct?

What is the HCF of $x^3+y^3+3xy-1$ and $(x+y{)}^4-1$?

What is the LCM of $x^4+x^2{y}^2+y^4$, $x^{3}y+y^4$ and $x^4{y}^2-x^3{y}^3$?

A number $N$ is such that when divided by 4, 6, 7 or 9, it leaves 3 as remainder. What is the smallest 4-digit number that satisfies this property?

A plank of wood 4.25 m long and 3.4 m wide is to be cut into square pieces of equal size. How many square pieces of largest size can be cut from the plank, if no wastage is allowed?

What is the HCF of $x^4-13x^2{y}^2-300y^4$, $x^3-4x^{2}y-4x{y}^2-5y^3$ and $x^3-125y^3$?

If HCF of 768 and $x^6{y}^2$ is $32xy$ for natural numbers $x\ge 2$, $y\ge 2$, then what is the value of $(x+y)$?

Let $N$ be a 5-digit number. When $N$ is divided by 6, 12, 15, 24 it leaves respectively 2, 8, 11, 20 as remainders. What is the greatest value of $N$?