Tag Archives: Divisibility

C(n,2) – C(k,2) is divisible by an odd prime p

      Given two integers       and an odd prime     Prove that if     is divisible by   ,   it must be divisible also by                 … Continue reading

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Numbers 561 and 1105

    The two smallest composite numbers for which     and   are     and   .     N.B.   It is not known whether there exist infinitely many composite numbers for which     and   … Continue reading

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Divisibility: 11|(a + 13b) and 13|(a + 11b)

  Definition : If     and     are integers (with     not zero), we say   divides   if there is an integer     such that   . we write,     means     divides … Continue reading

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Divisibility puzzle : a*b*c | (a + b + c)^n, n = 7,13,21,31

  Definition : If   a   and   b   are integers (with   a   not zero),   we say   a divides   b if there is an integer   c   such that   . we … Continue reading

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Divisibility| Inserting zeros between two consecutive digits of an integer

  7   divides   42     Claim: If   ,   where   p   is a prime number (in the example   p = 7   and   N = 42) If     is the number … Continue reading

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Puzzle| Repunit numbers – divisibility

  Repunit numbers are the positive integers of the form of     (n digits)                     is a prime number.           Prove that     is divisible by   41   if and only if   … Continue reading

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Numbers where the n-th digits is divisible by the n-th prime

    means a divides b.   The first 10 prime numbers:   2, 3, 5, 7, 11, 13, 17, 19, 23, 29 Consider the numbers:   6300846559   and   8757193191                 … Continue reading

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Palindromes that become square when multiplied (or divided) by 2,3,5

  Part #1 : Palindromes that become square when multiplied (or divided) by 7                                                                                                                                                                                                                                                                                                                                                                                     … Continue reading

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Consecutive integers divisible by consecutive odd numbers – Part 2

  Older post: Consecutive integers divisible by consecutive odd numbers – Part 1   22   consecutive integers which are divisible by   5, 7, 9, 11, …, 47   respectively:   23   consecutive integers which are divisible by … Continue reading

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Consecutive integers divisible by consecutive odd numbers

    2 consecutive integers which are divisible by   5, 7,   respectively:     3 consecutive integers which are divisible by   5, 7, 9,   respectively:     4 consecutive integers which are divisible by   5, … Continue reading

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