Divisibility, congruence classes, prime numbers
(1) a) Find the smallest positive integer in the given set:
{48u + 72v|u, v ∈ Z};
{171u + 25v|u, v ∈ Z}.
b) Find all integer solutions to the equations 17x + 95y = 1 and 23x + 101y = 5.
(2) a) Prove that for every n, the greatest common divisor of n + 1 and n2 − n + 1 is 1 or 3.
b) What are the possible greatest common divisors of n4 + 16 and n + 2?
(3) a) If 3|(a2 + b2 ), prove that 3|a and 3|b.
b) If 5|(a2 + b2 + c2 ), prove that 5|a or 5|b or 5|c.
(4) Let a, b and c be integer numbers such that 6 divides a + b + c. Show that 6 also divides
a5 + b3 + c.
(5) Let a, b, c be natural numbers. If 2a3 − 3a2 b + 2b3 is divisible by 5, show that a and b are
divisible by 5.
(6) Show that the sum of squares of 5 consecutive numbers cannot be a perfect square.
(7) If p is prime, prove that p2 + 2 is composite.
(8) Find all prime numbers p such that p2 − 2, 2p2 − 1 and 3p2 + 4 are also primes. (Hint:
think of the possible remainders of p when divided by 7).
(9) a) Show that there are infinitely many prime numbers of the form 3k + 2, where k is
integer.
b) Same for 4k + 3, where k is integer.
c) Same for 6k + 5, where k is integer.
(10) Show that among 12 consecutive natural numbers there exists one smaller than the sum
of its divisors.
(11) Given any 40 numbers, such that none of them is divisible by 5. Show that the sum of
their 40-th powers is divisible by 5.
(12) If a, b, c, d are integers not divisible by 5, show that a4 + 2b4 + 3c4 + 4d4 is divisible by 5.
(13) (IMO 2003) Let p be a prime number. Prove that there exists a prime number q such
that for any integer n, the number np − p is not divisible by q.
(14) The least common multiple of nonzero integers a and b is the smallest positive integer
m such that a|m and b|m. It is usually denoted m = [a, b]. Let d = (a, b) be the g.c.d.
If a = pr11 pr22 · · · prkk and b = ps11 ps22 · · · pskk , where p1 , p2 , ..., pk are distinct positive primes
and each ri , si ≥ 0, then
a) (a, b) = pn1 1 pn2 2 · · · pnk k where for each i, ni = minimum of ri , si .
b) [a, b] = pt11 pt22 · · · ptkk where for each i, ti = maximum of ri , si .
ab
c) Prove that [a, b] = (a,b)
.
d) How many divisors does a have? Write down the sum of all the divisors in terms of
pi -s and ri -s.
(15) Decompose each of the following numbers into a product of primes:
a) 217 + 25 − 1;
b) 212 − 25 − 1;
c) 213 − 24 + 1.
(16) Find the smallest positive integer congruent to 2 mod 3, to 4 mod 5, and to 5 mod 7.
a) Find all solutions of the equation x6 + x + 3 ≡ 0 mod 5.
b) Find all solutions of the equation x6 + x + 3 ≡ 0 mod 25.
c) Find all solutions of the equation x6 + x + 3 ≡ 0 mod 75.
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Hints
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(15)
Euclid’s algorithm for finding the gcd of two numbers.
Euclid’s algorithm.
Check possible values of a2 , b2 mod 3;respectively mod 5.
Check possible values of a, b, c mod 6.
b 6≡ 0 mod 5. Use Euclid’s algorithm to prove that there exists x such that 5x ≡ 1 mod
5. Using the given equation, write an equation for ax and show that it does not have
integer solutions.
(n − 2)2 + (n − 1)2 + n2 + (n + 1)2 + (n + 2)2 = 5(n2 + 2). Look at n2 + 2 mod 5.
Think of the possible remainders of p when divided by 6.
Think of the possible remainders of p when divided by 7.
b) Assume p1 , p2 , ..., pl are all the primes of that form except for 3. Argue that 4p1 p2 ...pl +3
has a prime divisor of the required type.
One of the 12 consecutive numbers has to be divisible by 12, be it 12k. Write the sum of
its known divisors.
Fermat’s little theorem.
Fermat’s little theorem.
Choose q a prime factor of pp−1 + pp−2 + ... + p + 1 such that q 6≡ 1 (mod p2 ) (first show
that such a q exists). Use Fermat’s little theorem.
d) Count all possible products of pi -s with powers at most ni .
Work mod 3, mod 5, mod 7 etc.