Download Homework 5 (=Exam Practice)

Theory of Numbers, Summer 2014
Jenya Soprunova
Homework 5 (=Exam Practice)
The exam will cover everything we did so far: induction, binomial theorem, divisibility, division
algorithm, gcd, euclidean algorithm, fundamental theorem of arithmetic, prime numbers, congruences, divisibility tests, chinese remainder theorem, Fermat’s theorem, pseudoprimes, Wilson’s theorem. Please review the proofs of infinitude of primes, Euclid’s lemma (if a prime p divides the
product of two integers ab, it divides at least one of a and b; if gcd(a,b)=1
and each of these
√
integers divides an integer c then their ab divides c), the irrationality of 2. Memorize all basic
definitions and be able to use them. Memorize the statements of Fermat’s Theorem and Wilson’s
Theorem. You need to be able to compute the gcd and express it as a linear combination using the
Euclidean algorithm. You need to able to solve a system of congruences. There will be a problem
on mathematical induction. You will need to be able to use divisibility tests. Here are some practice
questions. You do not need to write them up and submit, they are for you to prepare for the exam.
Problem 1. Prove that for all values of n
2 · 6 · 10 · (4n − 2) =
(2n)!
n!
Problem 2. For m ∈ N show that
2m
2
m
=2
+ m2 .
2
Problem 3. Show that the cube of any integer is of the form 9k, 9k + 1, or 9k + 8.
Problem 4. Show that 3a2 + 2 is never a perfect square.
Problem 5. Show that an integer all of whose digits are equal to 1 is a perfect square if and only
if it is equal to 1. (Let N = 111 . . . 1, what is N equal to modulo 4? What can a perfect square be
equal to modulo 4?)
Problem 6. Use mathematical induction to show that
5 | 33n+1 + 2n+1
Problem 7. Let gcd(a, b) = 1, d | ac and d | bc, then d | c. (Do not use Fundamental Theorem
of Arithmetic.)
Problem 8. Use the Euclidean algorithm to find the gcd of 119 and 272 and to express it as a
linear combination of 119 and 272.
Problem 9. Let gcd(a, b) = 1. Show that gcd(a + b, a − b) is either 1 or 2.
Problem 10. Let p ≥ 5 be prime, show that p2 +2 is composite. (Hint: What can be the remainder
of p after it is divided by 6?)
Problem 11. For n > 1, show that n! is not a square of an integer.
Problem 12. Find p such that both p and p2 + 8 are prime.
Problem 13. Show that 7 | 111333 + 333111 .
Problem 14. For an integer a, show that a4 ≡ 0 or 1 mod 5. What can be the units digit of a4 ?
(For each of your answers give an example to show that it actually occurs.)
Problem 15. Show that a number N whose tens digit is a and units digit is b is congruent modulo
4 to the two-digit number ab. Conclude that N is divisible by 4 if and only if ab is divisible by 4.
Problem 16. Let N = x14y38z . GIven that 396| N , find all possible values for N .
2
Problem 17. Solve a system of congruences: 5x ≡ 3 mod 11, 7x ≡ 2 mod 11.
Problem 18. Show that 17 | 11115 − 5.
Problem 19. Prove that a5 and a have the same units digit.
Problem 20. Find the remainder after 6 · 27! is divided by 31.