Divisibility

  • 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.
  • 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?

  • Fac-Finding
    problem

    Fac-finding

    Age
    14 to 16
    Challenge level
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    Lyndon chose this as one of his favourite problems. It is accessible but needs some careful analysis of what is included and what is not. A systematic approach is really helpful.
  • Three times Seven
    problem

    Three times seven

    Age
    11 to 14
    Challenge level
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    A three digit number abc is always divisible by 7 when 2a+3b+c is divisible by 7. Why?
  • Just Repeat
    problem

    Just repeat

    Age
    11 to 14
    Challenge level
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    Think of any three-digit number. Repeat the digits. The 6-digit number that you end up with is divisible by 91. Is this a coincidence?
  • 396
    problem

    396

    Age
    14 to 16
    Challenge level
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    The four digits 5, 6, 7 and 8 are put at random in the spaces of the number : 3 _ 1 _ 4 _ 0 _ 9 2 Calculate the probability that the answer will be a multiple of 396.
  • Remainders
    problem

    Remainders

    Age
    7 to 14
    Challenge level
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    I'm thinking of a number. My number is both a multiple of 5 and a multiple of 6. What could my number be?

  • Remainder
    problem

    Remainder

    Age
    11 to 14
    Challenge level
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    What is the remainder when 2^2002 is divided by 7? What happens with different powers of 2?
  • Divisively so
    problem

    Divisively so

    Age
    11 to 14
    Challenge level
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    How many numbers less than 1000 are NOT divisible by either: a) 2 or 5; or b) 2, 5 or 7?
  • Powerful factorial
    problem

    Powerful factorial

    Age
    11 to 14
    Challenge level
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    6! = 6 x 5 x 4 x 3 x 2 x 1. The highest power of 2 that divides exactly into 6! is 4 since (6!) / (2^4 ) = 45. What is the highest power of two that divides exactly into 100!?