4.3 Divisibility. Warm Up . a. 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.
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a. 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?
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
To prove p is true is equivalent to assume p is FALSE and then arrive to a False known fact
Prove by using contrapositive
The smallest positive factor different than one of an integer is a prime number.
The 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.
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.
Representing numbers in the form 7q+r
-25=7(-4)+3 14=7(2)+0 19=7(2)+5