Download Exercises - UVic Math

Number Theory Questions
1. Find the base 16 representation of 262 139.
2. Is it true that (121)b is a square in any base b? Why or why not?
3. Find a base b such that (122)b = 101.
4. Find x if (123)4 = x5 .
5. Show that a number in base 3 is even if and only of the sum of its digits is even.
In which other bases is this true?
6. Let a, b, c, d ∈ Z. Prove that if a|b and c|d, then ac|bd.
7. Let a, b, d ∈ Z. Prove that if d|a and d|b, then d2 |ab.
8. Let a, b, c, d ∈ Z, and suppose that a + b = c. Prove that if d divides and two
of a, b, c, then it also divides the third of these.
9. Prove that if a|b, then ab |b. Make sure you are using the definition of the
statement “a divides b”.
10. Explain why the Fundamental Theorem of Arithmetic implies that there are no
positive integers a and b such that 2a = 3b .
11. For a positive integer n, define n factorial to be the integer n(n − 1)(n − 2) · · · 1.
(a) Suppose 1 ≤ k ≤ n. What are the quotient and remainder when N = n!+1
is divided by k?
(b) Explain why part (a) implies that N has a prime divisor greater than n.
(c) Explain why part (b) implies that there are infinitely many prime numbers.
(Note that of there are only finitely many prime numbers, then there is a
largest prime.)
12. Find the prime factorization of N = 2·3·5·7·11+1. (Note that N is composite.)
13. Find the prime factorization of 16!. (Note that it is not necessary to compute
16! first.
14. Let q1 , q2 and q3 be different primes. Prove that if p is prime and p|q1 q2 q3 , then
p ∈ {q1 , q2 , q3 }.
15. Prove that the integer n is a perfect square if and only if every exponent in
its prime factorization is even. State the corresponding result for prefect k-th
powers, k ≥ 2.
16. Find the smallest natural number that is divisible by 2 and by 3, and which is
simultaneously the fourth power of an integer, and the sixth power of an integer.
Answer the same question when 2 is replaced by 4.
17. Let a, b, d ∈ Z. Suppose that d|ab. Is it always true that d|a or d|b? Give a
proof or counterexample, as appropriate.
18. Let n be a positive integer. Is it possible for a prime p to divide both n and
n + 1?
19. Let n be a positive integer. Prove that log2 (n) is irrational unless n is a power
of 2.
20. Suppose gcd (a, b) = 4. What are the possibilities for gcd (9a, b)?
21. Use the prime factorizations of 25 · (24)3 and 10! to find the prime factorizations
of gcd (25 · (24)3 , 10!) and lcm(25 · (24)3 , 10!).
22. How many positive divisors does 25 34 53 have?
23. Suppose that a and b are integers such that ab = −27 38 52 76 and gcd(a, b) =
23 34 5. Is it possible that a = 25 34 5? Why or why not? What is lcm(a, b)?
24. Let a and b be positive integers such that gcd (a, b) = 1. Prove that lcm(a, b) =
ab.
25. Use the Euclidean Algorithm to find gcd (8288, 15392). Use your work to find
(a) lcm(8288, 15392);
(b) Integers x and y such that 8288x + 15392y = gcd (8288, 15392);
(c) For k ∈ Z, integers xk and yk such that 8288xk +15392yk = k·gcd (8288, 15392).
26. Let d = gcd (a, b). Prove that gcd (a/d, b/d) = 1.
27. Suppose c is a common divisor of a and b, that is c | a and c | b. Prove that
c | gcd (a, b).
28. Suppose that there are integers x and y so that ax + by = 2. Suppose d is an
odd divisor of a such that d|bc. Prove that d|c.
29. Suppose there are integers x and x such that ax + by = 12. What are the
possibilities for gcd (a, b)? Why?
30. Prove that gcd (n, n + 1) = 1 for all n ∈ Z. What are the possibilities for
gcd (n, n + 2), gcd (n, n + 3) and gcd (n, n + 4)?
31. Let a ∈ Z and k ∈ N. Prove that one of the numbers a, a + 1, . . . , a + (k − 1) is
divisible by k.
(a) Let n ∈ Z and m ∈ N. The least residue of n modulo m is the unique
integer among 0, 1, . . . , m − 1 to which n is congruent modulo m. For
k ∈ Z, which numbers can be the least residue of k 2 modulo 4?
(b) Prove that no integer which is congruent to 3 modulo 4 can be written as
a sum of two squares. That is, if n ≡ 3 (mod 4), then there are no integers
x and y such that n = x2 + y 2 . (Hint: the contrapositive.)
32. Give an example to show that the statement
if ac ≡ bc (mod m,) then a ≡ b (mod m)
is false in general. Now show that the statement is true if, in addition, we know
that gcd (c, m) = 1.
33. Show that every odd prime is congruent to 1 or 3 modulo 4. If p > 3 is prime,
to what can p be congruent to modulo 6?
34. Use congruences to find the last digit of 4343 , and the last two digits of 747 .
35. (a) Let b > 1 be an integer, and n = (dk dk−1 . . . d1 d0 )b . Show that (b−1) | n ⇔
(b − 1) | d0 + d1 + d2 + · · · + dk .
(b) Let n = (dk dk−1 . . . d1 d0 )10 . Show that 11 | n ⇔ 11 | d0 − d1 + d2 − · · · +
(−1)k dk . (Hint: 10 ≡ (−1) (mod 11)).
(c) Part (a) is a generalization of the familiar statement that 9|n if and only if
it divides the sum of the digits of n. State a similar generalization of the
result in part (b).
36. (a) Given that k ≡ 2 (mod 4), determine the remainder when 5k+13 is divided
by 4.
(b) Given that k ≡ 1 (mod 4), determine the remainder when 7k 333 + 11 is
divided by 4.
37. Use congruences to prove that 13 | 19n − 6n for any n ≥ 0. More generally,
prove that if a and b are integers, then d = a − b divides an − bn for any n ≥ 0.