Modular arithmetic properties

Questions proving properties or identities using modular arithmetic, including showing divisibility via congruences.

7 questions · Standard +0.8

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OCR Further Additional Pure AS 2023 June Q1
3 marks Moderate -0.8
1
  1. Express 205 in the form \(7 q + r\) for positive integers \(q\) and \(r\), with \(0 \leqslant r < 7\).
  2. Given that \(7 \mid ( 205 \times 8666 )\), use the result of part (a) to justify that \(7 \mid 8666\).
OCR Further Additional Pure 2020 November Q7
10 marks Challenging +1.2
7 Throughout this question, \(n\) is a positive integer.
  1. Explain why \(n ^ { 5 } \equiv n ( \bmod 5 )\).
  2. By proving that \(n ^ { 5 } \equiv n ( \bmod 2 )\), show that \(n ^ { 5 } \equiv n ( \bmod 10 )\).
    1. Prove that \(n ^ { 5 } - n\) is divisible by 30 for all positive integers \(n\).
    2. Is there an integer \(N\), greater than 30 , such that \(n ^ { 5 } - n\) is divisible by \(N\) for all positive integers \(n\) ? Justify your answer.
Edexcel FP2 AS 2019 June Q2
7 marks Standard +0.8
  1. (i) Determine all the possible integers \(a\), where \(a > 3\), such that
$$15 \equiv 3 \bmod a$$ (ii) Show that if \(p\) is prime, \(x\) is an integer and \(x ^ { 2 } \equiv 1 \bmod p\) then either $$x \equiv 1 \bmod p \quad \text { or } \quad x \equiv - 1 \bmod p$$ (iii) A company has \(\pounds 13940220\) to share between 11 charities. Without performing any division and showing all your working, decide if it is possible to share this money equally between the 11 charities.
Edexcel FP2 AS 2023 June Q5
8 marks Standard +0.3
    1. Making your reasoning clear and using modulo arithmetic, show that
$$214 ^ { 6 } \text { is divisible by } 8$$ (ii) The following 7-digit number has four unknown digits $$a 5 \square b \square a b 0$$ Given that the number is divisible by 11
  1. determine the value of the digit \(a\). Given that the number is also divisible by 3
  2. determine the possible values of the digit \(b\).
Edexcel FP2 2020 June Q8
12 marks Challenging +1.8
  1. The four digit number \(n = a b c d\) satisfies the following properties:
    (1) \(n \equiv 3 ( \bmod 7 )\)
    (2) \(n\) is divisible by 9
    (3) the first two digits have the same sum as the last two digits
    (4) the digit \(b\) is smaller than any other digit
    (5) the digit \(c\) is even
    1. Use property (1) to explain why \(6 a + 2 b + 3 c + d \equiv 3 ( \bmod 7 )\)
    2. Use properties (2), (3) and (4) to show that \(a + b = 9\)
    3. Deduce that \(c \equiv 5 ( a - 1 ) ( \bmod 7 )\)
    4. Hence determine the number \(n\), verifying that it is unique. You must make your reasoning clear.
OCR Further Additional Pure AS 2024 June Q6
9 marks
6 For positive integers \(n\), let \(f ( n ) = 1 + 2 ^ { n } + 4 ^ { n }\).
    1. Given that \(n\) is a multiple of 3 , but not of 9 , use the division algorithm to write down the two possible forms that \(n\) can take.
    2. Show that when \(n\) is a multiple of 3 , but not of 9 , \(f ( n )\) is a multiple of 73 .
  2. Determine the value of \(\mathrm { f } ( n )\), modulo 73 , in the case when \(n\) is a multiple of 9 .
OCR Further Additional Pure AS 2021 November Q7
10 marks Challenging +1.2
7
  1. Let \(f ( n ) = 2 ^ { 4 n + 3 } + 3 ^ { 3 n + 1 }\). Use arithmetic modulo 11 to prove that \(\mathrm { f } ( n ) \equiv 0 ( \bmod 11 )\) for all integers \(n \geqslant 0\).
  2. Use the standard test for divisibility by 11 to prove the following statements.
    1. \(10 ^ { 33 } + 1\) is divisible by 11
    2. \(10 ^ { 33 } + 1\) is divisible by 121