Modular arithmetic

  • Check Codes
    problem

    Check codes

    Age
    14 to 16
    Challenge level
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    Details are given of how check codes are constructed (using modulus arithmetic for passports, bank accounts, credit cards, ISBN book numbers, and so on. A list of codes is given and you have to check if they are valid identification numbers?
  • Obviously?
    problem

    Obviously?

    Age
    14 to 18
    Challenge level
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    Find the values of n for which 1^n + 8^n - 3^n - 6^n is divisible by 6.
  • Pythagoras mod 5
    problem

    Pythagoras mod 5

    Age
    16 to 18
    Challenge level
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    Prove that for every right angled triangle which has sides with integer lengths: (1) the area of the triangle is even and (2) the length of one of the sides is divisible by 5.
  • Rational Round
    problem

    Rational round

    Age
    16 to 18
    Challenge level
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    Show that there are infinitely many rational points on the unit circle and no rational points on the circle x^2+y^2=3.
  • Dirisibly Yours
    problem

    Dirisibly yours

    Age
    16 to 18
    Challenge level
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    Find and explain a short and neat proof that 5^(2n+1) + 11^(2n+1) + 17^(2n+1) is divisible by 33 for every non negative integer n.
  • A One in Seven Chance
    problem

    A one in seven chance

    Age
    11 to 14
    Challenge level
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    What is the remainder when 2^{164}is divided by 7?
  • Take Three From Five
    problem

    Take three from five

    Age
    11 to 16
    Challenge level
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    Caroline and James pick sets of five numbers. Charlie tries to find three that add together to make a multiple of three. Can they stop him?

  • The Best Card Trick?
    problem

    The best card trick?

    Age
    11 to 16
    Challenge level
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    Time for a little mathemagic! Choose any five cards from a pack and show four of them to your partner. How can they work out the fifth?
  • Where can we visit?
    problem

    Where can we visit?

    Age
    11 to 14
    Challenge level
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    Charlie and Abi put a counter on 42. They wondered if they could visit all the other numbers on their 1-100 board, moving the counter using just these two operations: x2 and -5. What do you think?
  • Two Much
    problem

    Two much

    Age
    11 to 14
    Challenge level
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    Explain why the arithmetic sequence 1, 14, 27, 40, ... contains many terms of the form 222...2 where only the digit 2 appears.