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Contents
- Whole Topic Notes
- 1A The Division Algorithm
- 1B The Euclidean Algorithm
- 1C Modular Arithmetic
- 1D Divisibility Tests
- 1E Solving Congruence Equations*
- 1F Fermat’s Little Theorem*
- 1G Combinatorics*
- Whole Topic Summary Resources (Including Past Paper Questions)
Whole Topic Notes
Full Notes to Accompany Examples
Dr Frost
- Dr Frost Further Pure 2 Number Theory Section
1A The Division Algorithm
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1B The Euclidean Algorithm
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- 785d: Use the Euclidean algorithm to find the Greatest Common Divisor (Higher Common Factor) of any two integers.
- 785e: Use the result a|b and a|c \implies a|(bx+cy) to find common factors of two expressions in the form pn + q and rn + s for any integer n.
- 786a: Determine integers x and y such that ax + by = \textrm{gcd}(a,b)
- 786b: Understand the significance of solutions x and y existing to an equation given in the form ax + by = 1
- 785: Exam Practice: Division theorem and Euclidean algorithm
- 786: Exam Practice: Bezout's Identity
1C Modular Arithmetic
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- 787a: Determine if two integers a and b are congruent modulo k.
- 787b: List integers congruent to another.
- 787c: Understand the significance of a \equiv 0 \,(\textrm{mod} \, k)
- 787d: Use the law of exponentiation for modulo arithmetic to find the remainder when large powers are divided by an integer.
- 787e: Determine the last digit of a large power.
- 787f: Reason about last digits of expressions containing factorials.
- 787: Exam Practice: Modular arithmetic
1D Divisibility Tests
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- 163a: Know if a number is divisible by 2, 3, 4, 5, 6, 8, 9 or 10.
- 163b: Know if a number is divisible by 11.
- 163c: Determine an unknown digit to make a number divisible by 3, 4, 6, 8 or 9.
- 163d: Know if a number is divisible by a large number by combining divisibility rules.
- 787g: Reason about the digits of a number in a divisibility problem using modulo arithmetic.
- 163: Exam Practice: Divisibility laws from 3 to 11 and dealing with larger divisors
1E Solving Congruence Equations*
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- 791a: Solve a congruence equation given in the form x \equiv a \,(\textrm{mod}\, k), giving the least residue.
- 791c: Determine the number of solutions of a congruence equation given in the form ax \equiv b \,(\textrm{mod}\, k)
- 791d: Solve a congruence equation given in the form ax \equiv b \,(\textrm{mod}\, k) using simple cancelling.
- 791e: Determine if two integers are multiplicative inverses modulo k.
- 791f: Determine a multiplicative inverse of an integer modulo k.
- 791g: Use multiplicative inverses to solve a congruence equation given in the form ax \equiv b \,(\textrm{mod}\, k)
- 791h: Use solutions to Bezout's identity to solve a congruence equation given in the form ax \equiv b \,(\textrm{mod}\, k)
- 791: Exam Practice: Solving congruence equations
1F Fermat’s Little Theorem*
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1G Combinatorics*
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Key Skills
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- 637a: Use the factorial function to get the number of arrangements of objects.
- 637b: Use the factorial function to get the number of arrangements of multiple groups.
- 637c: Determine the number of arrangements of letters in a word with repeated letters.
- 637d: Use a single application of a permutation function.
- 637e: Use a single application of a choose function.
- 637f: Determine the number of choices when picking multiple items from two or more groups.
- 637h: Determine the number of arrangements when a group of objects needs to be kept together.
- 637k: Determine a probability involving the choose function.
- 637n: Determine the number of subsets of a set.
- 637: Exam Practice: Permutations and Combinations
Worksheets
- Mathshelper (Jon Stone)