Chapter 10: Further Pure 2: Number Theory

A-level Further MathsFurther Pure 1 & 27 topicsHigher

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Further Pure 2: Number Theory: Edexcel Further Maths Optional Long-Answer Questions

Examiner-style long-answer practice with automatic marking, improvement advice and ProofCoach tracking.

4 questions32 marks64 min

standard / hard / synoptic

8 marksfindstandardEdexcel

FP10-1 Find the required result for Further Pure 2: Number Theory. Given Find the remainder when

7^{222}

is divided by 13, and prove that

5^{6}

- 1 is divisible by 7. Use use modular arithmetic, Fermat's Little Theorem and residue cycles. (8 marks)

Define the variables, diagram labels, distribution parameters or algorithm table columns needed.
Substitute the concrete values from the question before simplifying.
Show the intermediate line that earns the main method mark.
Complete the calculation, table trace or proof and state the final result.

0 characters

Write your answer first, then mark it.

8 marksshow thathardEdexcel

FP10-2 Show that

7^{12}

is congruent to 1 modulo 13, so

7^{222}

is congruent to

7^{6}

for Further Pure 2: Number Theory. Start from Find the remainder when

7^{222}

is divided by 13, and prove that

5^{6}

- 1 is divisible by 7. (8 marks)

Define the variables, diagram labels, distribution parameters or algorithm table columns needed.
Substitute the concrete values from the question before simplifying.
Show the intermediate line that earns the main method mark.
Complete the calculation, table trace or proof and state the final result.

0 characters

Write your answer first, then mark it.

8 marksdeterminehardEdexcel

FP10-3 Determine the requested value in this structured Further Pure 2: Number Theory problem: simplify the final residue and justify each congruence. Use the same data: Find the remainder when

7^{222}

is divided by 13, and prove that

5^{6}

- 1 is divisible by 7. (8 marks)

Define the variables, diagram labels, distribution parameters or algorithm table columns needed.
Substitute the concrete values from the question before simplifying.
Show the intermediate line that earns the main method mark.
Complete the calculation, table trace or proof and state the final result.

0 characters

Write your answer first, then mark it.

8 markshencesynopticEdexcel

FP10-4 Hence use the result

7^{12}

is congruent to 1 modulo 13, so

7^{222}

is congruent to

7^{6}

to complete the extension for Further Pure 2: Number Theory: simplify the final residue and justify each congruence. (8 marks)

Define the variables, diagram labels, distribution parameters or algorithm table columns needed.
Substitute the concrete values from the question before simplifying.
Show the intermediate line that earns the main method mark.
Complete the calculation, table trace or proof and state the final result.

0 characters

Write your answer first, then mark it.

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9. Further Pure 1: Reducible Differential Equations

11. Further Pure 2: Groups

Key learning objectives

Understand the key principles behind Further Pure 2: Number Theory.
Apply these ideas to exam-style questions.
Recall the essential definitions and key terms.