The marbles used to tile the floor are square marbles.

Therefore, $\text{the length of the marble} = \text{width of the marble}.$

As we have to use whole number of marbles, the side of the square should a factor of both 5 m 78 cm and 3m 74. And it should be the highest factor of 5 m 78 cm and 3m 74.

5 m 78 cm = 578 cm and 3 m 74 cm = 374 cm.
The HCF of 578 and 374 = 34.

Hence, the side of the square is 34.

The number of such square marbles required,

$= \dfrac{578×374}{34×34}$
$= 17×11=$ 187 marbles

Q4.

What number should be subtracted from $x^3+ 4x^2- 7x + 12$, if it is to be perfectly divisible by $x + 3$?

According to remainder theorem when $dfrac{f(x)}{x+a}$, then the remainder is $f(-a)$.

In this case, as $x + 3$ divides $x^3 + 4x^2 - 7x + 12 – k$ perfectly ($k$ being the number to be subtracted), the remainder is 0 when the value of $x$ is substituted by -3.

i.e., $(-3)^3 + 4(-3)^2 - 7(-3) + 12 - k = 0$

or $-27 + 36 + 21 + 12 = k$

or $k =$ 42

Q5.

Find the remainder when $2^{89}$ is divided by $89$?