Elementary number theory

Question 1

what is the process for proof by induction

Answer 1

try n = 1
try k
using k, prove k + 1

Question 2

n is a natural number, with n > 1, then either n is prime, or is a product of a (finite) sequence of primes.

Answer 2

n is a natural number, with n > 1, then either n is prime, or is a product of a (finite) sequence of primes.

Question 3

Theorem 3.14 — Fundamental Theorem of Arithmetic. Let n have the prime factorizations
n=p1···pr =q1···qs.
Then every prime occurs equally often in both factorizations (and so r = s).

Answer 3

Theorem 3.14 — Fundamental Theorem of Arithmetic. Let n have the prime factorizations
n=p1···pr =q1···qs.
Then every prime occurs equally often in both factorizations (and so r = s).

Question 4

Definition 3.15 For an integer n ≥ 1, we say that integers a and b are congruent modulo n, whenever (a − b)/n is an integer.

Answer 4

Definition 3.15 For an integer n ≥ 1, we say that integers a and b are congruent modulo n, whenever (a − b)/n is an integer.

Question 5

a ≡ a (mod n) for all a (so congruence modulo n is reflexive),
* ifa≡b(modn)thenb≡a(modn)(socongruencemodulon is symmetric),
* ifa≡b(modn)andb≡c(modn)thena≡c(modn)(so congruence modulo n is transitive).

Answer 5

a ≡ a (mod n) for all a (so congruence modulo n is reflexive),
* ifa≡b(modn)thenb≡a(modn)(socongruencemodulon is symmetric),
* ifa≡b(modn)andb≡c(modn)thena≡c(modn)(so congruence modulo n is transitive).