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EPGY
Math Olympiad
Math Olympiad Problem Solving
Stanford University EPGY Summer Institutes 2008
Problem Set: Introduction to Number Theory
1. Find all positive integers n for which
(n + 1) | (n2 + 1).
2. If 7 | (3x + 2) prove that 7 | (15x2 − 11x − 14.).
3. Show that the square of any integer is of the form 3k or 3k + 1.
4. Show that if the sides of a right triangle are all integers, then 3 divides the
length of one of the sides.
5. Suppose that n is an integer such that 5|(n + 2). Which of the following are
divisible by 5?
(a)
(b)
(c)
(d)
n2 − 4
n2 + 8n + 7
n4 − 1
n2 − 2n
6. Find all integers n ≥ 1 so that n3 −1 is prime. Hint: n3 −1 = (n2 +n+1)(n−1).
7. (a) Prove that the product of three consecutive integers is divisible by 6.
(b) Prove that the product of four consecutive integers is divisible by 24.
(c) Prove that the product of n consecutive integers is divisible by n!.
8. Prove that n5 − 5n3 + 4n is divisible by 120 for all integers n ≥ 1.
9. Find all integers n ≥ 1 so that n4 + 4 is prime.
10. Find all integers n ≥ 1 so that n4 + 4n is prime.
11. Prove that the square of any integer of the form 5k + 1 is of the same form.
12. Prove that 3 is not a divisor of n2 + 1 for all integers n ≥ 1.
13. A prime triplet is a triple of numbers of the form (p, p + 2, p + 4), for which p,
p + 2, and p + 4 are all prime. For example, (3, 5, 7) is a prime triplet. Prove
that (3, 5, 7) is the only prime triplet.
14. Prove that if 3 | (a2 + b2 ), then 3 | a and 3 | b. Hint: If 3 - a and 3 - b, what are
the possible remainders upon division by 3?
Summer 2008
1
Number Theory Problem Set
EPGY
Math Olympiad
15. Find the largest positive integer n such that
(n + 1)(n4 + 2n) + 3(n3 + 57)
be divisible by n2 + 2.
16. Show that if n is a positive integer such that 2n + 1 is a square, then n + 1 is
the sum of two consecutive squares.
17. Prove that there are infinitely many primes of the form 6n − 1.
18. Prove that there are infinitely many primes p such that p − 2 is not prime.
19. Show that there are no three consecutive odd integers such that each is the sum
of two squares greater than zero.
20. Prove that if n is an even natural number, then the number 13n + 6 is divisible
by 7.
21. Let n > 1 be a positive integer. Prove that if one of the numbers 2n − 1, 2n + 1
is prime, then the other is composite.
22. Prove that there are infinitely many integers n such that 4n2 + 1 is divisible by
both 13 and 5.
23. Prove that the product of four consecutive natural numbers is never a perfect
square.
Summer 2008
2
Number Theory Problem Set