1. A number $N$ is divided by $D$ resulted in quotient $q$ and remainder $r$. Then $N$ can be written as
a. $Dq - r$
b. $\dfrac{D}{q} - r$
c. $Dq + r$
d. $Dr + q$

Answer: D

Explanation:
$\quad\require{enclose}
\begin{array}{rll}
q &\\[0pt]
D \enclose{longdiv}{\;N}\kern-.2ex \\[0pt]
\underline{Dq}{\phantom{}} \\[0pt]
r\phantom{} \\[0pt]
\end{array}$
$\therefore$ $N-Dq = r$
$\therefore$ $N = Dq + r$ Note: The above result is very important and you should learn how to write a division in this format)

2. One to ten numbers are written side by side. If this number is divided by 8 then what is the remainder?
a. 5
b. 6
c. 7
d. 0

Answer: B

Explanation:
If one to ten numbers are written side by side, last three digits are $910$.
$\quad\require{enclose}
\begin{array}{rll}
113 &\\[0pt]
8 \enclose{longdiv}{910}\kern-.2ex \\[0pt]
\underline{8}{\phantom{00}} \\[0pt]
11\phantom{0} \\[0pt]
\underline {8}{\phantom{0}} \\[0pt]
30\phantom{} \\[0pt]
\underline{24}\phantom{} \\[0pt]
\boxed6\phantom{}
\end{array}$

3. What is the value of $k$ if $8975k$ is divisible by 7
a. 11
b. 16
c. 7
d. 9

Answer: B

Explanation:
$\quad\require{enclose}
\begin{array}{rll}
128\,\, &\\[0pt]
7 \enclose{longdiv}{8975k}\kern-.2ex \\[0pt]
\underline{7}{\phantom{0000}} \\[0pt]
19\phantom{000} \\[0pt]
\underline {14}{\phantom{000}} \\[0pt]
57\phantom{00} \\[0pt]
\underline{56}\phantom{00} \\[0pt]
1k\phantom{0} \\[0pt]
\underline{1k}\phantom{0} \\[0pt]
0\phantom{0}
\end{array}$
$\therefore$ $1k$ has to be a multiple of $7$.
The only value possible is $14$.
So $k = 4$

4. A number when divided by 69 leaves a remainder of 50. Find the remainder when it is divided by 23.
a. 10
b. 20
c. 6
d. 4

Answer: D

Explanation:
The number $N$ can be written as $N = 69q + 50$.
If the above expression is divided by $23$, $69q$ gives a remainder $0$ as it is a multiple of $23$.
When $50$ is divided by $23$, the remainder is $4$.

5. A number when divided by a 23 leaves a remainder of 5. Find the remainder when the original number is multiplied by 20 and then divided by the same divisor.
a. 8
b. 10
c. 19
d. 20

Answer: A

Explanation:
The number $N$ can be written as $N = 23q + 5$.
If the above expression is multiplied by $20$, Then $20N = 460q + 100$.
When $460q$ is divided by 23, it gives a remainder $0$ as it is a multiple of $23$.
When $100$ is divided by $23$, the remainder is $8$.

6. A number when divided by a 23 leaves a remainder of 5. Find the remainder when square of the number is divided by the same divisor.
a. 1
b. 2
c. 4
d. 8

Answer: B

Explanation:
The number $N$ can be written as $N = 23q + 5$.
If the above expression is squared, then $N^2 = {\left( {23q + 5} \right)^2}$ = ${23^2} \times {q^2} + 2 \times 23q \times 5 + {25}$
${23^2} \times {q^2}$, $2 \times 23q \times 5$ are multiples of 23, thus give remainder $0$.
When $25$ is divided by $23$, the remainder is $2$.