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14 Divisors

What is the smallest number with exactly 14 divisors?

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Repeaters

Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 594594). Explain why whatever digits you choose the number will always be divisible by 7, 11 and 13.

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Oh! Hidden Inside?

Find the number which has 8 divisors, such that the product of the divisors is 331776.

Square Routes

Stage: 3 Challenge Level: Challenge Level:3 Challenge Level:3 Challenge Level:3

  1. How many 4 digit squares are composed of even numerals?
    These square numbers must be between 2000 and 8888 which means that they are squares of numbers between 45 and 94. Moreover the units digit is even. Ong Xing Cong found the following solutions:

    68 x 68 = 4624

    78 x 78 = 6084

    80 x 80 = 6400

    92 x 92 = 8464

  2. What 4 digit square when reversed becomes the square of another number and why?

    Allan has cracked yet another tough nut but others may have further comments to make about this one.

    The numbers are the squares of 33 and 99, they are 1089 and 9801.

    First, I noticed that multiples of 11, when reversed, are still multiples of 11, like 143, which becomes 341, when reversed, which is divisible by 11.
    Next, I found that some multiples of 11 x 11, which is 121, when reversed, are also divisible by 121.
    Then I found that multiples of 9, when reversed, are still divisible by 9.
    The number 1089, which is the lowest common multiple of 121 and 9, when multipled by 9, gives the number 9801.
    The square root of 1089 is 33, and the square root of 9801 is 99.