Divisibility

  • Eminit
    problem

    Eminit

    Age
    11 to 14
    Challenge level
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    The number 8888...88M9999...99 is divisible by 7 and it starts with the digit 8 repeated 50 times and ends with the digit 9 repeated 50 times. What is the value of the digit M?
  • American Billions
    problem

    American billions

    Age
    11 to 14
    Challenge level
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    Play the divisibility game to create numbers in which the first two digits make a number divisible by 2, the first three digits make a number divisible by 3...
  • Adding in Rows
    problem

    Adding in rows

    Age
    11 to 14
    Challenge level
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    List any 3 numbers. It is always possible to find a subset of adjacent numbers that add up to a multiple of 3. Can you explain why and prove it?
  • Why 24?
    problem

    Why 24?

    Age
    14 to 16
    Challenge level
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    Take any prime number greater than 3 , square it and subtract one. Working on the building blocks will help you to explain what is special about your results.
  • Factoring factorials
    problem

    Factoring factorials

    Age
    11 to 14
    Challenge level
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    Find the highest power of 11 that will divide into 1000! exactly.
  • Neighbours
    problem

    Neighbours

    Age
    7 to 11
    Challenge level
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    In a square in which the houses are evenly spaced, numbers 3 and 10 are opposite each other. What is the smallest and what is the largest possible number of houses in the square?
  • Book Codes
    problem

    Book codes

    Age
    7 to 11
    Challenge level
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    Look on the back of any modern book and you will find an ISBN code. Take this code and calculate this sum in the way shown. Can you see what the answers always have in common?
  • Gaxinta
    problem

    Gaxinta

    Age
    11 to 14
    Challenge level
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    A number N is divisible by 10, 90, 98 and 882 but it is NOT divisible by 50 or 270 or 686 or 1764. It is also known that N is a factor of 9261000. What is N?
  • DigAt
    problem

    Digat

    Age
    11 to 14
    Challenge level
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    What is the value of the digit A in the sum below: [3(230 + A)]^2 = 49280A
  • AB Search
    problem

    AB search

    Age
    11 to 14
    Challenge level
    filled star filled star empty star

    The five digit number A679B, in base ten, is divisible by 72. What are the values of A and B?