DP Mathematics HL Questionbank

• Syllabus

10.2

Sub sections and their related questions

\(\left. a \right|b \Rightarrow b = na\) for some \(n \in \mathbb{Z}\) .

• 18M.3dm.hl.TZ0.4a: Show that...
• 18M.3dm.hl.TZ0.4b.i: State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for odd positive integers \(k\).
• 18M.3dm.hl.TZ0.4b.ii: State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for even positive...

The theorem \(\left. a \right|b\) and \(a\left| {c \Rightarrow a} \right|\left( {bx \pm cy} \right)\) where \(x,y \in \mathbb{Z}\) .

• 18M.3dm.hl.TZ0.4a: Show that...
• 18M.3dm.hl.TZ0.4b.i: State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for odd positive integers \(k\).
• 18M.3dm.hl.TZ0.4b.ii: State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for even positive...

Division and Euclidean algorithms.

• 12M.3dm.hl.TZ0.1a: Use the Euclidean algorithm to express gcd (123, 2347) in the form 123p + 2347q, where...
• 12N.3dm.hl.TZ0.3a: Using the Euclidean algorithm, find h .
• 12N.3dm.hl.TZ0.3b: Hence find integers A and B such that 861A + 957B = h .
• 08M.3dm.hl.TZ1.1: Use the Euclidean Algorithm to find the greatest common divisor of 7854 and 3315. Hence state...
• 08M.3dm.hl.TZ2.1: (a)     Use the Euclidean algorithm to find the gcd of 324 and 129. (b)     Hence show that...
• 08N.3dm.hl.TZ0.1b: (i)     Using the Euclidean algorithm, find the greatest common divisor, d , of 901 and...
• 11M.3dm.hl.TZ0.1a: Use the Euclidean algorithm to find the greatest common divisor of the numbers 56 and 315.
• 09M.3dm.hl.TZ0.2: (a)     Use the Euclidean algorithm to find gcd(\(12\,306\), 2976) . (b)     Hence give the...
• SPNone.3dm.hl.TZ0.1a: Use the Euclidean algorithm to find the greatest common divisor of 259 and 581.
• 13M.3dm.hl.TZ0.1a: Using the Euclidean algorithm, show that \(\gcd (99,{\text{ }}332) = 1\).
• 11N.3dm.hl.TZ0.2a: Use the Euclidean algorithm to find \(\gcd (752,{\text{ }}352)\).
• 14N.3dm.hl.TZ0.1a: Find the values of \(f(3)\), \(f(4)\) and \(f(5)\).
• 14N.3dm.hl.TZ0.1b: Use the Euclidean algorithm to find (i)     \(\gcd \left( {f(3),{\text{ }}f(4)} \right)\); (ii)...
• 14N.3dm.hl.TZ0.1d: By factorizing \(f(n)\) explain why it is always exactly divisible by \(6\).
• 14N.3dm.hl.TZ0.1e: Determine the values of \(n\) for which \(f(n)\) is exactly divisible by \(60\).
• 15M.3dm.hl.TZ0.5b: Use the Fundamental theorem of arithmetic, applied to \(5577\) and \(99\,099\), to calculate...
• 16M.3dm.hl.TZ0.1a: Use the Euclidean algorithm to show that 1463 and 389 are relatively prime.
• 18M.3dm.hl.TZ0.4a: Show that...
• 18M.3dm.hl.TZ0.4b.i: State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for odd positive integers \(k\).
• 18M.3dm.hl.TZ0.4b.ii: State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for even positive...

The greatest common divisor, gcd(\(a\),\(b\)), and the least common multiple, lcm(\(a\),\(b\)), of integers \(a\) and \(b\).

• 12M.3dm.hl.TZ0.1a: Use the Euclidean algorithm to express gcd (123, 2347) in the form 123p + 2347q, where...
• 11M.3dm.hl.TZ0.1a: Use the Euclidean algorithm to find the greatest common divisor of the numbers 56 and 315.
• 11M.3dm.hl.TZ0.5b: (i)     State the fundamental theorem of arithmetic. (ii)     The positive integers M and N have...
• 14M.3dm.hl.TZ0.2a: Consider the integers \(a = 871\) and \(b= 1157\), given in base \(10\). (i)     Express...
• 14N.3dm.hl.TZ0.1b: Use the Euclidean algorithm to find (i)     \(\gcd \left( {f(3),{\text{ }}f(4)} \right)\); (ii)...
• 15M.3dm.hl.TZ0.5b: Use the Fundamental theorem of arithmetic, applied to \(5577\) and \(99\,099\), to calculate...
• 15M.3dm.hl.TZ0.5c: Prove that \(\gcd (n,{\text{ }}m) \times {\text{lcm}}(n,{\text{ }}m) = n \times m\) for all...
• 18M.3dm.hl.TZ0.4a: Show that...
• 18M.3dm.hl.TZ0.4b.i: State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for odd positive integers \(k\).
• 18M.3dm.hl.TZ0.4b.ii: State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for even positive...

Prime numbers; relatively prime numbers and the fundamental theorem of arithmetic.

• 08N.3dm.hl.TZ0.3a: Write 57128 as a product of primes.
• 11M.3dm.hl.TZ0.5a: Explaining your method fully, determine whether or not 1189 is a prime number.
• 11M.3dm.hl.TZ0.5b: (i)     State the fundamental theorem of arithmetic. (ii)     The positive integers M and N have...
• 14M.3dm.hl.TZ0.2d: Consider the set \(P\) of numbers of the form \({n^2} - n + 41,{\text{ }}n \in \mathbb{N}\). (i)...
• 13N.3dm.hl.TZ0.5a: Show that \(30\) is a factor of \({n^5} - n\) for all \(n \in \mathbb{N}\).
• 15M.3dm.hl.TZ0.5a: State the Fundamental theorem of arithmetic for positive whole numbers greater than \(1\).
• 15M.3dm.hl.TZ0.5b: Use the Fundamental theorem of arithmetic, applied to \(5577\) and \(99\,099\), to calculate...
• 18M.3dm.hl.TZ0.4a: Show that...
• 18M.3dm.hl.TZ0.4b.i: State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for odd positive integers \(k\).
• 18M.3dm.hl.TZ0.4b.ii: State the value of \({\text{gcd}}\left( {4k + 2,\,3k + 1} \right)\) for even positive...