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250 Problems In Elementary Number Theory

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Problems 1-40 : Divisibility of Numbers

These pages contain problems and solutions from Section 1 - "Divisibility of Numbers" of the book "250 Problems In Elementary Number Theory"

Problems: [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [20a] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

Problem 1

Find all positive integers n such that n^2+1 is divisible by n+1.

[Hint] -- [Solution] -- [Glossary]

Problem 2

Find all integers x != 3 such that x-3 | x^3 - 3.

[Hint] -- [Solution] -- [Glossary]

Problem 3

Prove that there exists infinitely many positive integers n such that 4*(n^2)+1 is divisible both by 5 and 13.

[Hint] -- [Solution] -- [Glossary]

Problem 4

Prove that for positive integers n we have 169 | 3^(3*n+3) - 26*n - 7.

[Hint] -- [Solution] -- [Glossary]

Problem 5

Prove that 19 | 2^(2^(6*k+2))+3 for k=0, 1, 2, 'ellipsis

[Hint] -- [Solution] -- [Glossary]

Problem 6

Prove the theorem, due to Kraïtchik, asserting that 13 | 2^70 + 2^30.

[Hint] -- [Solution] -- [Glossary]

Problem 7

Prove that 11 * 31 * 61 | 20^15-1.

[Hint] -- [Solution] -- [Glossary]

Problem 8

Prove that for positive integer m and a > 1 we have gcd ( (a^m-1) / (a-1) , a-1) = gcd ( a-1,m ).

[Hint] -- [Solution] -- [Glossary]

Problem 9

Prove that for every positive integer n the number 3 * (1^5 + 2^5 + 'ellipsis + n^5) is divisible by 1^3 + 2^3 + 'ellipsis + n^3.

[Hint] -- [Solution] -- [Glossary]

Problem 10

Find all integers n > 1 such that 1^n + 2^n + 'ellipsis + (n-1)^n is divisible by n.

[Hint] -- [Solution] -- [Glossary]

Problem 11

For positive integer n, find which of the two numbers a subscript n = 2^(2*n+1) - 2^(n+1) +1 and b subscript n = 2^(2*n+1) + 2^(n+1) +1 is divisible by 5 and which is not.

[Hint] -- [Solution] -- [Glossary]

Problem 12

Prove that for every positive integer n there exists a positive integer x such that each of the terms of the infinite sequence x+1, x^x+1, x^(x^x)+1, 'ellipsis is divisible by n

[Hint] -- [Solution] -- [Glossary]

Problem 13

Prove that there exist infinitely many positive integers n such that for every even x none of the terms of the sequence x^x+1, x^(x^x)+1, x^(x^(x^x))+1, 'ellipsis is divisible by n.

[Hint] -- [Solution] -- [Glossary]

Problem 14

Prove that for positive integer n we have n^2 | (n+1)^n - 1.

[Hint] -- [Solution] -- [Glossary]

Problem 15

Prove that for positive integer n we have (2^n-1)^2 | 2^((2^n-1)*n) - 1.

[Hint] -- [Solution] -- [Glossary]

Problem 16

Prove that there exist infinitely many positive integers n such that n | 2^n+1 ; find all such prime numbers.

[Hint] -- [Solution] -- [Glossary]

Problem 17

[Hint] -- [Solution] -- [Glossary]

Problem 18

[Hint] -- [Solution] -- [Glossary]

Problem 19

Find all positive integers a for which a^10+1 is divisble by 10.

[Hint] -- [Solution] -- [Glossary]

Problem 20

[Hint] -- [Solution] -- [Glossary]

Problem 20a

Prove that there exist infinitely many positive integers n such that n|2^n+1.
This problem is a subset of problem 16.

[Hint] -- [Solution] -- [Glossary]

Problem 21

Find all odd n such that n|3^n+1.

[Hint] -- [Solution] -- [Glossary]

Problem 22

Find all positive integers n such that 3|n^2^n+1.

[Hint] -- [Solution] -- [Glossary]

Problem 23

Prove that for every off prime p there exist infinitely many positive integers n such that p|n*2^n+1.

[Hint] -- [Solution] -- [Glossary]

Problem 24

[Hint] -- [Solution] -- [Glossary]

Problem 25

[Hint] -- [Solution] -- [Glossary]

Problem 26

[Hint] -- [Solution] -- [Glossary]

Problem 27

[Hint] -- [Solution] -- [Glossary]

Problem 28

[Hint] -- [Solution] -- [Glossary]

Problem 29

[Hint] -- [Solution] -- [Glossary]

Problem 30

[Hint] -- [Solution] -- [Glossary]

Problem 31

[Hint] -- [Solution] -- [Glossary]

Problem 32

[Hint] -- [Solution] -- [Glossary]

Problem 33

[Hint] -- [Solution] -- [Glossary]

Problem 34

Prove that if for integers a and b we have 7|a^2+b^2 then 7|a and 7|b.

[Hint] -- [Solution] -- [Glossary]

Problem 35

[Hint] -- [Solution] -- [Glossary]

Problem 36

[Hint] -- [Solution] -- [Glossary]

Problem 37

[Hint] -- [Solution] -- [Glossary]

Problem 38

[Hint] -- [Solution] -- [Glossary]

Problem 39

Prove that if a, b, c are any integers, and n is an integer () .gt 3, then there exists an integer k such that none of the numbers k+a, k+b, k+c is divisibly by n.

[Hint] -- [Solution] -- [Glossary]

Problem 40

[Hint] -- [Solution] -- [Glossary]

	This page last changed: 15 May 2001
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