A regular polygon is a plane shape that is enclosed by $3$ or more lines of equal lengths, and it is always symmetric, having all the interior angles equal to each other; examples include equilateral triangle, square, regular pentagon, regular hexagon and so on.

Verma has $120$ identical matchsticks, and he wants to form a regular polygon, how many distinct polygons can he create using all the matchsticks together?

Which of the following is necessarily a factor of

$(1279+279-79-9{)}^{127}-\left({1279}^{127}+{279}^{127}-{79}^{127}-9^{127}\right)$

Find the number of non-negative integral values of x that satisfy the following equation, where $\lfloor \mathrm{x}\rfloor$ denotes the greatest integer less than or equal to x.

$⌊\frac{\mathrm{x}}{9}⌋=⌊\frac{{\mathrm{x}}^2}{99}⌋$

If $\lfloor \mathrm{x}\rfloor$ denotes the greatest integer less than or equal to x, which one of the following could be a suitable value of x for which each term of the following expression is expressed in its simplest form?

$\frac{⌊\mathrm{x}⌋+3}{4},\frac{⌊\mathrm{x}⌋+8}{5},\frac{⌊\mathrm{x}⌋+15}{6},\frac{\lfloor \mathrm{x}\rfloor +24}{7},\dots .\frac{\lfloor \mathrm{x}\rfloor +168}{15},\frac{\lfloor \mathrm{x}\rfloor +195}{16}$