DP Mathematics HL Questionbank

• Syllabus

10.4

Sub sections and their related questions

Modular arithmetic.

• 12M.3dm.hl.TZ0.1b: Find the least positive solution of $123x \equiv 1(\bmod 2347)$ .
• 12M.3dm.hl.TZ0.1c: Find the general solution of $123z \equiv 5(\bmod 2347)$ .
• 12M.3dm.hl.TZ0.1d: State the solution set of $123y \equiv 1(\bmod 2346)$ .
• 12M.3dm.hl.TZ0.5a: Use the result $2003 = 6 \times 333 + 5$ and Fermat’s little theorem to show that...
• 12M.3dm.hl.TZ0.5b: Find ${2^{2003}}(\bmod 11)$ and ${2^{2003}}(\bmod 13)$.
• 12M.3dm.hl.TZ0.5c: Use the Chinese remainder theorem, or otherwise, to evaluate ${2^{2003}}(\bmod 1001)$, noting...
• 12N.3dm.hl.TZ0.3c: Using part (b), solve $287w \equiv 2(\bmod 319$) , where...
• 08M.3dm.hl.TZ1.2a: Define what is meant by the statement...
• 08M.3dm.hl.TZ1.2b: Hence prove that if $x \equiv y(\bmod n)$ then ${x^2} \equiv {y^2}(\bmod n)$ .
• 08M.3dm.hl.TZ1.2c: Determine whether or not ${x^2} \equiv {y^2}(\bmod n)$ implies that $x \equiv y(\bmod n)$ .
• 11M.3dm.hl.TZ0.1b: (i)     Find the general solution to the diophantine equation $56x + 315y = 21$. (ii)    ...
• 11M.3dm.hl.TZ0.3a: Given that a , $b \in \mathbb{N}$ and $c \in {\mathbb{Z}^ + }$, show that if...
• 11M.3dm.hl.TZ0.3b: Using mathematical induction, show that ${9^n} \equiv 1(\bmod 4)$ , for $n \in \mathbb{N}$ .
• 10M.3dm.hl.TZ0.5: Given that $a,{\text{ }}b,{\text{ }}c,{\text{ }}d \in \mathbb{Z}$, show...
• 11N.3dm.hl.TZ0.4: Anna is playing with some cars and divides them into three sets of equal size. However, when she...
• 14M.3dm.hl.TZ0.2c: Consider the simultaneous equations      $4x + y + 5z = a$      $2x + z = b$     ...
• 13N.3dm.hl.TZ0.5b: (i)     Show that ${3^{{3^m}}} \equiv 3({\text{mod}}4)$ for all $m \in \mathbb{N}$. (ii)    ...
• 14N.3dm.hl.TZ0.4c: Using your answers to parts (a) and (b) find the remainder when ${41^{82}}$ is divided by $55$.
• 15N.3dm.hl.TZ0.3b: Hence, or otherwise, find the remainder when ${1982^{1982}}$ is divided by $37$.
• 15N.3dm.hl.TZ0.5a: Given a sequence of non negative integers $\{ {a_r}\}$ show that (i)    ...
• 15N.3dm.hl.TZ0.5b: Hence determine whether the base $3$ number $22010112200201$ is divisible by $8$.
• 17N.3dm.hl.TZ0.4a: With reference to the integers 5, 8 and 3, state why the Chinese remainder theorem guarantees a...
• 17N.3dm.hl.TZ0.4b: Hence or otherwise, find the general solution to the above system of linear congruences.
• 18M.3dm.hl.TZ0.2b.ii: Hence solve the linear congruence $5x \equiv 7\left( {{\text{mod}}\,13} \right)$.

The solution of linear congruences.

• 12M.3dm.hl.TZ0.1b: Find the least positive solution of $123x \equiv 1(\bmod 2347)$ .
• 12M.3dm.hl.TZ0.1c: Find the general solution of $123z \equiv 5(\bmod 2347)$ .
• 12M.3dm.hl.TZ0.1d: State the solution set of $123y \equiv 1(\bmod 2346)$ .
• 12M.3dm.hl.TZ0.5c: Use the Chinese remainder theorem, or otherwise, to evaluate ${2^{2003}}(\bmod 1001)$, noting...
• 12N.3dm.hl.TZ0.3c: Using part (b), solve $287w \equiv 2(\bmod 319$) , where...
• 08N.3dm.hl.TZ0.1c: In each of the following cases find the solutions, if any, of the given linear congruence. (i)  ...
• 13M.3dm.hl.TZ0.1b: (i)     Find the general solution to the diophantine equation $332x - 99y = 1$. (ii)    ...
• 18M.3dm.hl.TZ0.2b.ii: Hence solve the linear congruence $5x \equiv 7\left( {{\text{mod}}\,13} \right)$.

Solution of simultaneous linear congruences (Chinese remainder theorem).

• 12M.3dm.hl.TZ0.5c: Use the Chinese remainder theorem, or otherwise, to evaluate ${2^{2003}}(\bmod 1001)$, noting...
• 08M.3dm.hl.TZ2.3a: (i)     Given that $a \equiv d(\bmod n)$ and $b \equiv c(\bmod n)$ prove...
• 09M.3dm.hl.TZ0.4: Two mathematicians are planning their wedding celebration and are trying to arrange the seating...
• SPNone.3dm.hl.TZ0.3b: Find all the integers between 100 and 200 satisfying the simultaneous congruences...
• 10M.3dm.hl.TZ0.1b: Find the general solution to the simultaneous congruences \[x \equiv 3(\bmod...
• 10N.3dm.hl.TZ0.2: (a)     Find the general solution for the following system of congruences.     ...
• 14N.3dm.hl.TZ0.4a: Solve, by any method, the following system of linear congruences      \(x \equiv 9(\bmod...