4.3 Divisibility

Warm Upa. Is 21 divisible by 3?

b. Does 5 divide 40?

c. Does 7 | 42?

d. Is 32 a multiple of −16?

e. Is 6 a factor of 53?

f. Is 7 a factor of −7?

Basic Important Results

• Prove that for all integers a, b, and c, if a | b and b | c, then a | c.

• Is the following statement true or false? For all integers a and b, if a | b and b | a then a = b.

Basic Important Results

If a, b are positive integers and a is a factor of b, then a≤b

Since b=ak for some k≥1, then ak≥a, so b≥a

• The only divisors of 1 are 1 and -1

Proofs by Contrapositive and by Contradiction

To prove p is true is equivalent to assume p is FALSE and then arrive to a False known fact

Proofs

Prove directly:• The square of any even number is even• The square of any odd number is odd

Prove by using contrapositive• If the square of an integer is odd, the integer is odd• If the square of an integer is even, the integer is

even

Contrapositive

The smallest positive factor different than one of an integer is a prime number.

• Provide examples to convince yourself about it• Proof by contradiction.

The smallest positive divisor of an integer is a prime number

Unique Factorization TheoremThe most comprehensive statement about divisibility of integers is contained in the unique factorization of integers theorem.

Because of its importance, this theorem is also called the fundamental theorem of arithmetic.

The unique factorization of integers theorem says that any integer greater than 1 either is prime or can be written as a product of prime numbers in a way that is unique except, perhaps, for the order in which the primes are written.

Square root of 2 is irrational

• (By contradiction) Assume it is rational and arrive to a result that is false.

Quotient-Remainder Theorem

When any integer n is divided by any positive integer d, the result is a quotient q and a nonnegative remainder r that is smaller than d.

Quotient-Remainder Theorem

Representing numbers in the form 7q+r

7(-3)7(-4) 7(-2) 7(0)7(-1) 7(1) 7(2) 7(3)

-25 14 19

-25=7(-4)+3 14=7(2)+0 19=7(2)+5