Number-Theory-1 Exercises en
Number-Theory-1 Exercises en
Number-Theory-1 Exercises en
1 Divisibility
Beginner
1.1 Show that 900 divides 10!.
1.2 The product of two numbers, neither of which is divisible by 10, is 1000. Find the sum of the
numbers.
Intermediate
1.4 Show that:
(a) 5 · 17 | 52 · 17 + 3 · 5 · 9 + 5 · 3 · 8
(b) n(n + m) | 3mn2 + amn2 + 3n3 + an3
1.5 Find three three-digit positive integers, using nine dierent digits, so that the product ends in
four zeros.
1.6 (a) Find all positive integers that have exactly 41 divisors, and are divisible by 41.
(b) Find all positive integers that have exactly 42 divisors, and are divisible by 42.
Olympiad
1.7 Find all positive integers n such that n + 1 | n2 + 1.
1.8 Prove: for each positive integer n there exist n consecutive integers such that none of them is
prime.
1.9 Prove that there are innitely many positive integers n such that 2n is a square, 3n is a third
power and 5n is a fth power.
1
2 gcd and lcm
Beginner
2.1 (IMO 59) Show that the following fraction is in lowest terms.
21n + 4
14n + 3
Intermediate
2.3 Every positive integers greater than 6 is the of two coprime positive integers greater than 1.
2.4 We call positive integers a and b friends if a · b is a square. Show that if a and b are friends, then
a and lcm(a, b) are friends.
Olympiad
2.5 Let m and n be two positive integers which sum to a prime number. Show that m and n are
coprime. Seien m und n zwei natürliche Zahlen, deren Summe eine Primzahl ist. Zeige, dass m
und n teilerfremd sind.
2.6 (Canada 97) Find the number of pairs of positive integers (x, y) with x ≤ y which satisfy the
following equalites:
gcd(x, y) = 5! and lcm(x, y) = 50!
3 Upper bounds
Beginner
3.1 We call a rectangle nice if its sides are integer lengths and the area and circumference are integers
as well. Determine all nice rectangles.
2
Intermediate
3.3 We call a cuboid nice if its sides are integer lengths and the volume and surface are integers as
well. Determine all nice cuboids.
Olympiad
3.6 Show that the equation
y 2 = x(x + 1)(x + 2)(x + 3)
has no solution in the positive integers.
3.8 (IMO 98) Determine all pairs (a, b) of positive integers such that ab2 + b + 7 divides a2 b + a + b.
Questions from past olympiads are excellent preparation materials: they are of course at the correct
level of diculty, and additionally, all solutions can be found on www.imosuisse.ch. But make sure
to give the problems a good try before looking up the solutions!
4.1 (Preliminary round 2012, 1.) Determine all pairs (m, n) of positive integers such that mn
divides (m + 1)(n + 2).
4.2 (Preliminary round 2004, 1.) Find all positive integers a,b and n such that the following
equation holds:
a! + b! = 2n
4.3 (Preliminary round 2005, 3.) Let m and n be coprime positive integers. Show that m3 +
mn + n3 and mn(m + n) are coprime.
4.4 (Preliminary round 2011, 2.) Find all positive integers n such that n3 is the product of three
positive divisors of n.
3
4.5 (Preliminary round 2006, 1.) Determine all triples (p, q, r) of prime numbers such that
|p − q|, |q − r|, |r − p|
4.6 (Preliminary round 2008, 4.) Determine all positive integers n such that the number of
positive divisors of n are equal to the third smallest positive divisor of n.
4.7 (Preliminary round 2013, 4.) Determine all pairs (m, n) of positive integers such that
(m + 1)! + (n + 1)! = m2 n2