DP Mathematics HL Questionbank

• Syllabus

10.6

Sub sections and their related questions

Fermat’s little theorem.

• 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 ${2^{2003}}(\bmod 13)$.
• 12N.3dm.hl.TZ0.4a: State Fermat’s little theorem.
• 08M.3dm.hl.TZ2.3b: Show that ${x^{97}} - x + 1 \equiv 0(\bmod 97)$ has no solution.
• 08N.3dm.hl.TZ0.3c: Prove that $\left. {22} \right|{5^{11}} + {17^{11}}$.
• 09M.3dm.hl.TZ0.5: (a)     Using Fermat’s little theorem, show that, in base 10, the last digit of n is always equal...
• 09N.3dm.hl.TZ0.4b: The representation of the positive integer N in base p is denoted by ${(N)_p}$ . If...
• SPNone.3dm.hl.TZ0.3a: One version of Fermat’s little theorem states that, under certain conditions,...
• 10M.3dm.hl.TZ0.1a: (i)     One version of Fermat’s little theorem states that, under certain...
• 10N.3dm.hl.TZ0.4: (a)     Write down Fermat’s little theorem. (b)     In base 5 the representation of a natural...
• 13M.3dm.hl.TZ0.5a: Show that $\sum\limits_{k = 1}^p {{k^p} \equiv 0(\bmod p)}$.
• 13M.3dm.hl.TZ0.5b: Given that $\sum\limits_{k = 1}^p {{k^{p - 1}} \equiv n(\bmod p)}$ where...
• 11N.3dm.hl.TZ0.5a: Use the above result to show that if p is prime then ${a^p} \equiv a(\bmod p)$ where a is any...
• 11N.3dm.hl.TZ0.5b: Show that ${2^{341}} \equiv 2(\bmod 341)$.
• 11N.3dm.hl.TZ0.5c: (i)     State the converse of the result in part (a). (ii)     Show that this converse is not true.
• 14N.3dm.hl.TZ0.1c: Explain why $f(n)$ is always exactly divisible by $5$.
• 14N.3dm.hl.TZ0.4b: Find the remainder when ${41^{82}}$ is divided by 11.
• 15N.3dm.hl.TZ0.3b: Hence, or otherwise, find the remainder when ${1982^{1982}}$ is divided by $37$.
• 17N.3dm.hl.TZ0.2b: For every prime number $p > 3$, show that $p|{u_{p - 1}}$.
• 18M.3dm.hl.TZ0.2a: State Fermat’s little theorem.
• 18M.3dm.hl.TZ0.2b.i: Use Fermat’s little theorem to show that $x \equiv {a^{p - 2}}b\left( {{\text{mod}}\,p} \right)$.