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 smallest integer whose first n multiples all contain a 3
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 Set {2,b,c,d,e} such that the product of any two of them increased by 1 is a square
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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
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
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
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
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
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
Numbers where the nth digits is divisible by the nth 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
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
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
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