4.1 Divisibility and Natural Numbers
Thinking Critically
4.1 Divisibility Of Natural Numbers
4.2 Tests for Divisibility
4.3 Greatest Common Divisors and Least
Common Multiples
If a and b are whole numbers with b ≠ 0 and there is a whole number q such that a = bq, we say that b divides a.
We also say that b is a factor of a or a divisor of a and that a is a multiple of b. If b divides a and b is less than a, it is called a proper divisor of a. A natural number that possesses exactly two different factors, itself and 1, is called a prime number.
A natural number that possesses more than two different factors is called a composite number. The number 1 is called a unit; it is neither prime nor composite.
THE FUNDAMENTAL THEOREM OF ARITHMETIC (Simple­Product Form)
Every natural number greater than 1 is a prime or can be expressed as a product of primes in one, and only one, way apart from the order of the prime factors.
NOTE: This is why we do not think of 1 as either a prime or composite number, to preserve uniqueness of the product.
4.2 Tests for Divisibility
The Number of Primes
There are infinitely many primes.
Prime Divisors of n
If n is composite, then there is a prime p such that p divides n and DIVISIBILITY OF SUMS AND DIFFERENCES
Let d, a, and b be natural numbers. Then if d divides both a and b, then it also divides their sum, a + b, and their difference, a – b.
TESTS FOR DIVISIBILITY
By 2:
A natural number is divisible by 2 exactly when its base ten units digit is 0, 2, 4, 6, or 8.
By 5:
A natural number is divisible by 5 exactly when its base ten units digit is 0 or 5.
By 10:
A natural number is divisible by 10 exactly when its base ten units digit is 0.
By 3:
A natural number is divisible by 3 if and only if the sum of its digits is divisible by 3.
By 9:
A natural number is divisible by 9 if and only if the sum of its digits is divisible by 9.
By 11:
A natural number is divisible by 11 exactly when the sum of its digits in the even and odd positions have a difference that is divisible by 11.
By 4:
A natural number is divisible by 4 when the number represented by its last two digits is divisible by 4.
By 8:
A natural number is divisible by 8 when the number represented by its last three digits is divisible by 8.
DIVISIBILITY BY PRODUCTS
Let a and b be divisors of a natural number n.
If a and b have no common divisor other than 1, then their product ab is also a divisor of n.
4.3 Greatest Common Divisors (GCDs)
and Least Common Multiples (LCMs)
Let a and b be whole numbers not both 0. The greatest natural number d that divides both a and b is called their greatest common divisor and we write If m and n are any two natural numbers, then
Let a and b be natural numbers. The least natural number m that is a multiple of both a and b is called their least common multiple, and we write