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Problems in Elementary Number Theory

Peter Vandendriessche

Hojoo Lee

July 11, 2007

God does arithmetic. C. F. Gauss

Chapter 1

Introduction

The heart of Mathematics is its problems. Paul Halmos

Number Theory is a beautiful branch of Mathematics. The purpose of this book is to present a collection of interesting problems in elementary Number Theory. Many of the problems are mathematical competition problems from all over the world like IMO, APMO, APMC, Putnam and many others. The book has a supporting website at

http://www.problem-solving.be/pen/

which has some extras to offer, including problem discussion and (where available) solutions, as well as some history on the book. If you like the book, you’ll probably like the website.

I would like to stress that this book is unfinished. Any and all feedback, especially about errors in the book (even minor typos), is appreciated. I also appreciate it if you tell me about any challenging, interesting, beautiful or historical problems in elementary number theory (by email or via the website) that you think might belong in the book. On the website you can also help me collecting solutions for the problems in the book (all available solutions will be on the website only). You can send all comments to both authors at

peter.vandendriessche at gmail.com and ultrametric at gmail.com

or (preferred) through the website.

The author is very grateful to Hojoo Lee, the previous author and founder of the book, for the great work put into PEN. The author also wishes to thank Orlando Doehring , who provided old IMO short-listed problems, Daniel Harrer for contributing many corrections and solutions to the problems and Arne Smeets, Ha Duy Hung , Tom Verhoeff , Tran Nam Dung for their nice problem proposals and comments.

Lastly, note that I will use the following notations in the book:

Z the set of integers, N the set of (strictly) positive integers, N 0 the set of nonnegative integers.

Enjoy your journey!

Chapter 2

Divisibility Theory

Why are numbers beautiful? It’s like asking why is Beethoven’s Ninth Symphony beautiful. If you don’t see why, someone can’t tell you. I know numbers are beautiful. If they aren’t beautiful, nothing is. Paul Erd¨os

A 1. Show that if x, y, z are positive integers, then (xy + 1)(yz + 1)(zx + 1) is a perfect

square if and only if xy + 1, yz + 1, zx + 1 are all perfect squares.

Kiran S. Kedlaya

A 2. Find infinitely many triples (a, b, c) of positive integers such that a, b, c are in arithmetic progression and such that ab + 1, bc + 1, and ca + 1 are perfect squares.

AMM, Problem 10622, M. N. Deshpande

A 3. Let a and b be positive integers such that ab + 1 divides a^2 + b^2. Show that

a^2 + b^2 ab + 1

is the square of an integer.

IMO 1988/

A 4. If a, b, c are positive integers such that

0 < a^2 + b^2 − abc ≤ c,

show that a^2 + b^2 − abc is a perfect square. 1

CRUX, Problem 1420, Shailesh Shirali

A 5. Let x and y be positive integers such that xy divides x^2 + y^2 + 1. Show that

x^2 + y^2 + 1 xy

(^1) This is a generalization of A3! Indeed, a (^2) + b (^2) − abc = c implies that a^2 +b^2 ab+1 =^ c^ ∈^ N.

A 6.

(a) Find infinitely many pairs of integers a and b with 1 < a < b, so that ab exactly divides a^2 + b^2 − 1.

(b) With a and b as above, what are the possible values of

a^2 + b^2 − 1 ab

CRUX, Problem 1746, K. Guy and Richard J.Nowakowki

A 7. Let n be a positive integer such that 2 + 2

28 n^2 + 1 is an integer. Show that 2 + 2

28 n^2 + 1 is the square of an integer. 1969 E¨otv¨os-K¨ursch´ak Mathematics Competition

A 8. The integers a and b have the property that for every nonnegative integer n the number of 2na + b is the square of an integer. Show that a = 0.

Poland 2001

A 9. Prove that among any ten consecutive positive integers at least one is relatively prime to the product of the others.

[IHH, pp. 211]

A 10. Let n be a positive integer with n ≥ 3. Show that

nn nn − nn n

is divisible by 1989.

[UmDz pp.13] Unused Problem for the Balkan MO

A 11. Let a, b, c, d be integers. Show that the product

(a − b)(a − c)(a − d)(b − c)(b − d)(c − d)

is divisible by 12.

Slovenia 1995

A 12. Let k, m, and n be natural numbers such that m + k + 1 is a prime greater than n + 1. Let cs = s(s + 1). Prove that the product

(cm+1 − ck)(cm+2 − ck) · · · (cm+n − ck)

is divisible by the product c 1 c 2 · · · cn.

Putnam 1972

A 13. Show that for all prime numbers p,

Q(p) =

p∏− 1

k=

k^2 k−p−^1

is an integer.

[GhEw pp.104]

A 24. Let p > 3 is a prime number and k = b 23 p c. Prove that ( p 1

p 2

p k

is divisible by p^2.

Putnam 1996

A 25. Show that

( 2 n n

| lcm(1, 2 , · · · , 2 n) for all positive integers n.

A 26. Let m and n be arbitrary non-negative integers. Prove that

(2m)!(2n)! m!n!(m + n)!

is an integer. 2

IMO 1972/

A 27. Show that the coefficients of a binomial expansion (a + b)n^ where n is a positive integer, are all odd, if and only if n is of the form 2k^ − 1 for some positive integer k.

A 28. Prove that the expression gcd(m, n) n

n m

is an integer for all pairs of positive integers (m, n) with n ≥ m ≥ 1.

Putnam 2000

A 29. For which positive integers k, is it true that there are infinitely many pairs of positive integers (m, n) such that (m + n − k)! m! n! is an integer?

AMM Problem E2623, Ivan Niven

A 30. Show that if n ≥ 6 is composite, then n divides (n − 1)!.

A 31. Show that there exist infinitely many positive integers n such that n^2 + 1 divides n!.

Kazakhstan 1998

A 32. Let a and b be natural numbers such that

a b

Prove that a is divisible by 1979. (^2) Note that 0! = 1.

IMO 1979/

A 33. Let a, b, x ∈ N with b > 1 and such that bn^ − 1 divides a. Show that in base b, the number a has at least n non-zero digits.

IMO Short List 1996

A 34. Let p 1 , p 2 , · · · , pn be distinct primes greater than 3. Show that

2 p^1 p^2 ···pn^ + 1

has at least 4n^ divisors.

IMO Short List 2002 N

A 35. Let p ≥ 5 be a prime number. Prove that there exists an integer a with 1 ≤ a ≤ p − 2 such that neither ap−^1 − 1 nor (a + 1)p−^1 − 1 is divisible by p^2.

IMO Short List 2001 N

A 36. Let n and q be integers with n ≥ 5, 2 ≤ q ≤ n. Prove that q − 1 divides

⌊ (^) (n−1)! q

Australia 2002

A 37. If n is a natural number, prove that the number (n + 1)(n + 2) · · · (n + 10) is not a perfect square.

Bosnia and Herzegovina 2002

A 38. Let p be a prime with p > 5, and let S = {p − n^2 |n ∈ N, n^2 < p}. Prove that S contains two elements a and b such that a|b and 1 < a < b.

MM, Problem 1438, David M. Bloom

A 39. Let n be a positive integer. Prove that the following two statements are equivalent.

• n is not divisible by 4
• There exist a, b ∈ Z such that a^2 + b^2 + 1 is divisible by n.

A 40. Determine the greatest common divisor of the elements of the set

{n^13 − n | n ∈ Z}.

[PJ pp.110] UC Berkeley Preliminary Exam 1990

A 41. Show that there are infinitely many composite numbers n such that 3n−^1 − 2 n−^1 is divisible by n.

[Ae pp.137]

A 42. Suppose that 2n^ + 1 is an odd prime for some positive integer n. Show that n must be a power of 2.

A 53. Suppose that x, y, and z are positive integers with xy = z^2 + 1. Prove that there exist integers a, b, c, and d such that x = a^2 + b^2 , y = c^2 + d^2 , and z = ac + bd.

Iran 2001

A 54. A natural number n is said to have the property P , if whenever n divides an^ − 1 for some integer a, n^2 also necessarily divides an^ − 1.

(a) Show that every prime number n has the property P.

(b) Show that there are infinitely many composite numbers n that possess the property P.

IMO ShortList 1993 IND

A 55. Show that for every natural number n the product ( 4 −

n

is an integer.

Czech and Slovak Mathematical Olympiad 1999

A 56. Let a, b, and c be integers such that a + b + c divides a^2 + b^2 + c^2. Prove that there are infinitely many positive integers n such that a + b + c divides an^ + bn^ + cn.

Romania 1987, L. Panaitopol

A 57. Prove that for every n ∈ N the following proposition holds: 7| 3 n^ + n^3 if and only if 7 | 3 nn^3 + 1.

Bulgaria 1995

A 58. Let k ≥ 14 be an integer, and let pk be the largest prime number which is strictly less than k. You may assume that pk ≥ 34 k. Let n be a composite integer. Prove that

(a) if n = 2pk, then n does not divide (n − k)!,

(b) if n > 2 pk, then n divides (n − k)!.

APMO 2003/

A 59. Suppose that n has (at least) two essentially distinct representations as a sum of two squares. Specifically, let n = s^2 + t^2 = u^2 + v^2 , where s ≥ t ≥ 0, u ≥ v ≥ 0, and s > u. Show that gcd(su − tv, n) is a proper divisor of n.

[AaJc, pp. 250]

A 60. Prove that there exist an infinite number of ordered pairs (a, b) of integers such that for every positive integer t, the number at + b is a triangular number if and only if t is a triangular number^5.

Putnam 1988/B (^5) The triangular numbers are the tn = n(n + 1)/2 with n ∈ { 0 , 1 , 2 ,... }.

A 61. For any positive integer n > 1, let p(n) be the greatest prime divisor of n. Prove that there are infinitely many positive integers n with

p(n) < p(n + 1) < p(n + 2).

Bulgaria 1995

A 62. Let p(n) be the greatest odd divisor of n. Prove that

1 2 n

∑^2 n

k=

p(k) k

Germany 1997

A 63. There is a large pile of cards. On each card one of the numbers 1, 2, · · · , n is written. It is known that the sum of all numbers of all the cards is equal to k · n! for some integer k. Prove that it is possible to arrange cards into k stacks so that the sum of numbers written on the cards in each stack is equal to n!.

[Tt] Tournament of the Towns 2002 Fall/A-Level

A 64. The last digit^6 of the number x^2 + xy + y^2 is zero (where x and y are positive integers). Prove that two last digits of this numbers are zeros.

[Tt] Tournament of the Towns 2002 Spring/O-Level

A 65. Clara computed the product of the first n positive integers and Valerid computed the product of the first m even positive integers, where m ≥ 2. They got the same answer. Prove that one of them had made a mistake.

[Tt] Tournament of the Towns 2001 Fall/O-Level

A 66. (Four Number Theorem) Let a, b, c, and d be positive integers such that ab = cd. Show that there exists positive integers p, q, r, s such that

a = pq, b = rs, c = ps, d = qr.

[PeJs, pp. 5]

A 67. Suppose that S = {a 1 , · · · , ar} is a set of positive integers, and let Sk denote the set of subsets of S with k elements. Show that

lcm(a 1 , · · · , ar) =

∏^ r

i=

s∈Si

gcd(s)((−1)

i) .

[Her, pp. 14]

A 69. Prove that if the odd prime p divides ab^ − 1, where a and b are positive integers, then p appears to the same power in the prime factorization of b(ad^ − 1), where d = gcd(b, p − 1).

MM, June 1986, Problem 1220, Gregg Partuno (^6) Base 10.

A 77. Find all positive integers, representable uniquely as

x^2 + y xy + 1

where x and y are positive integers.

Russia 2001

A 78. Determine all ordered pairs (m, n) of positive integers such that

n^3 + 1 mn − 1

is an integer.

IMO 1994/

A 79. Determine all pairs of integers (a, b) such that

a^2 2 ab^2 − b^3 + 1

is a positive integer.

IMO 2003/

A 80. Find all pairs of positive integers m, n ≥ 3 for which there exist infinitely many positive integers a such that am^ + a − 1 an^ + a^2 − 1

is itself an integer.

IMO 2002/

A 81. Determine all triples of positive integers (a, m, n) such that am^ + 1 divides (a + 1)n.

IMO Short List 2000 N

A 82. Which integers can be represented as

(x + y + z)^2 xyz

where x, y, and z are positive integers?

AMM, Problem 10382, Richard K. Guy

A 83. Find all n ∈ N such that b

nc divides n. [Tma pp. 73]

A 84. Determine all n ∈ N for which

• n is not the square of any integer,
• b

nc^3 divides n^2.

India 1989

A 85. Find all n ∈ N such that 2n−^1 divides n!.

[ElCr pp. 11]

A 86. Find all positive integers (x, n) such that xn^ + 2n^ + 1 divides xn+1^ + 2n+1^ + 1.

Romania 1998

A 87. Find all positive integers n such that 3n^ − 1 is divisible by 2n.

A 88. Find all positive integers n such that 9n^ − 1 is divisible by 7n.

A 89. Determine all pairs (a, b) of integers for which a^2 + b^2 + 3 is divisible by ab.

Turkey 1994

A 90. Determine all pairs (x, y) of positive integers with y|x^2 + 1 and x|y^3 + 1.

Mediterranean Mathematics Competition 2002

A 91. Determine all pairs (a, b) of positive integers such that ab^2 + b + 7 divides a^2 b + a + b.

IMO 1998/

A 92. Let a and b be positive integers. When a^2 + b^2 is divided by a + b, the quotient is q and the remainder is r. Find all pairs (a, b) such that q^2 + r = 1977.

IMO 1977/

A 93. Find the largest positive integer n such that n is divisible by all the positive integers less than 3

n. APMO 1998

A 94. Find all n ∈ N such that 3n^ − n is divisible by 17.

A 95. Suppose that a and b are natural numbers such that

p = b 4

2 a − b 2 a + b

is a prime number. What is the maximum possible value of p?

Iran 1998

A 96. Find all positive integers n that have exactly 16 positive integral divisors d 1 , d 2 · · · , d 16 such that 1 = d 1 < d 2 < · · · < d 16 = n, d 6 = 18, and d 9 − d 8 = 17.

Ireland 1998

A 97. Suppose that n is a positive integer and let

d 1 < d 2 < d 3 < d 4

be the four smallest positive integer divisors of n. Find all integers n such that

n = d 12 + d 22 + d 32 + d 42.

IMO Long List 1987

A 107. Find four positive integers, each not exceeding 70000 and each having more than 100 divisors.

IMO Short List 1986 P10 (NL1)

A 108. For each integer n > 1, let p(n) denote the largest prime factor of n. Determine all triples (x, y, z) of distinct positive integers satisfying

• x, y, z are in arithmetic progression,
• p(xyz) ≤ 3.

British Mathematical Olympiad 2003, 2-

A 109. Find all positive integers a and b such that

a^2 + b b^2 − a and

b^2 + a a^2 − b

are both integers.

APMO 2002/

A 110. For each positive integer n, write the sum

∑n m=1 1 /m^ in the form^ pn/qn, where^ pn and qn are relatively prime positive integers. Determine all n such that 5 does not divide qn.

Putnam 1997/B

A 111. Find all natural numbers n such that the number n(n + 1)(n + 2)(n + 3) has exactly three different prime divisors.

Spain 1993

A 112. Prove that there exist infinitely many pairs (a, b) of relatively prime positive integers such that a^2 − 5 b

and b^2 − 5 a are both positive integers.

Germany 2003

A 113. Find all triples (l, m, n) of distinct positive integers satisfying

gcd(l, m)^2 = l + m, gcd(m, n)^2 = m + n, and gcd(n, l)^2 = n + l.

Russia 1997

A 114. What is the greatest common divisor of the set of numbers

{ 16 n^ + 10n − 1 | n = 1, 2 , · · · }?

[EbMk, pp. 16]

A 115. Does there exist a 4-digit integer (in decimal form) such that no replacement of three of its digits by any other three gives a multiple of 1992?

[Ams, pp. 102], I. Selishev

A 116. What is the smallest positive integer that consists base 10 of each of the ten digits, each used exactly once, and is divisible by each of the digits 2 through 9?

[JDS, pp. 27]

A 117. Find the smallest positive integer n such that

21989 | mn^ − 1

for all odd positive integers m > 1.

[Rh2, pp. 98]

A 118. Determine the highest power of 1980 which divides

(1980n)! (n!)^1980

MM, Jan. 1981, Problem 1089, M. S. Klamkin

B 5. Let p be an odd prime. If g 1 , · · · , gφ(p−1) are the primitive roots (mod p) in the range 1 < g ≤ p − 1, prove that φ( ∑p−1)

i=

gi ≡ μ(p − 1) (mod p).

[Km, Problems Sheet 3-11]

B 6. Suppose that m does not have a primitive root. Show that

a

φ(m) (^2) ≡ 1 (mod m)

for every a relatively prime m.

[AaJc, pp. 178]

B 7. Suppose that p > 3 is prime. Prove that the products of the primitive roots of p between 1 and p − 1 is congruent to 1 modulo p. [AaJc, pp. 181]

B 8. Let p be a prime. Let g be a primitive root of modulo p. Prove that there is no k such that gk+2^ ≡ gk+1^ + 1 ≡ gk^ + 2 (mod p). [Her, pp. 99]

3.2 Quadratic Residues

C 1. Find all positive integers n that are quadratic residues modulo all primes greater than n. CRUX, Problem 2344, Murali Vajapeyam

C 2. The positive integers a and b are such that the numbers 15a + 16b and 16a − 15 b are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? IMO 1996/

C 3. Let p be an odd prime number. Show that the smallest positive quadratic nonresidue of p is smaller than √p + 1.

[IHH pp.147]

C 4. Let M be an integer, and let p be a prime with p > 25. Show that the set {M, M + 1 , · · · , M + 3b

pc − 1 } contains a quadratic non-residue to modulus p. [Imv, pp. 72]

C 5. Let p be an odd prime and let Zp denote (the field of) integers modulo p. How many elements are in the set {x^2 : x ∈ Zp} ∩ {y^2 + 1 : y ∈ Zp}? Putnam 1991/B

C 6. Let a, b, c be integers and let p be an odd prime with

p 6 |a and p 6 |b^2 − 4 ac.

Show that (^) p ∑

k=

ak^2 + bk + c p

a p

[Ab, pp. 34]

3.3 Congruences

D 1. If p is an odd prime, prove that ( k p

k p

(mod p).

[Tma, pp. 127]

D 2. Suppose that p is an odd prime. Prove that

∑^ p

j=

p j

p + j j

≡ 2 p^ + 1 (mod p^2 ).

Putnam 1991/B

D 3. Show that

(−1)

p− 21

p − 1 p− 1 2

≡ 4 p−^1 (mod p^3 )

for all prime numbers p with p ≥ 5.

Morley

D 4. Let n be a positive integer. Prove that n is prime if and only if ( n − 1 k

≡ (−1)k^ (mod n)

for all k ∈ { 0 , 1 , · · · , n − 1 }.

MM, Problem 1494, Emeric Deutsch and Ira M.Gessel

D 5. Prove that for n ≥ 2,

22 ···^2 ︸︷︷︸ n terms

···^2 ︸︷︷︸ n−1 terms

(mod n).

Putnam 1997/B

D 6. Show that, for any fixed integer n ≥ 1 , the sequence

2 , 22 , 22 2 , 22 22 , · · · (mod n)

is eventually constant.

USA 1991

D 7. Somebody incorrectly remembered Fermat’s little theorem as saying that the congruence an+1^ ≡ a (mod n) holds for all a if n is prime. Describe the set of integers n for which this property is in fact true.

[DZ] posed by Don Zagier at the St AndrewsColloquium 1996

D 8. Characterize the set of positive integers n such that, for all integers a, the sequence a, a^2 , a^3 , · · · is periodic modulo n.