Arithmetic
section 1.7
Factors, Primes, and prime factorization
Def
Natural numbers (positive integers) are 1, 2, 3, 4, 5, 6, ...
Def`
(Factor)
Let A be a natural number.
If p † q = A where p and q are natural numbers, then p and q are factors (divisors) of A.
Ex1,
Find all factors of 12.
Note that 12 = 1 † 12 = 2 † 6 = 3 † 4
Therefore, factors of 12 are 1, 2, 3, 4, 6, 12.
Ex2,
Find all factors of 50.
Ex3,
Find all factors of 120.
Ex4,
Find all factors of 19.
Def
Def
Prime (number) is a natural number which has exactly 2 factors. (Namely, 1 and the number itself)
A composite number is a natural number which has 3 or more different factors.
Note that 1 is not a prime and not a composite.
Ex5,
Is 23 a prime?
Ex6,
Is 51 a prime?
Ex7,
Is 91 a prime?
How do we check if a given natural number is a prime number or a composite number?
It is sufficient to check divisors which are prime numbers whose squares are less than the given number.
(Try to divide the number by every prime number whose square is less than the given number.)
Ex8,
Is 103 a prime number?
Ex9,
Is 111 a prime number?
Ex10, Is 143 a prime number?
Ex11, Is 169 a prime number?
List of prime numbers upto 101:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, ...
Def
Prime factorization of a natural number is the number expressed as a product of only prime numbers.
Ex12, Find the prime factorization of 12.
12 = 2 † 6 = 2 † 3 † 2 = 22 † 3
Ex13, Find the prime factorization of 120.
Ex14*, On a separate sheet of paper, find the prime factorization of every natural number from 1 through 100.
Ex15, Find the prime factorization of
(a) 87
(b) 131
(c) 133
(d) 301
Ex16, Find the prime factorization of
(a) 2457
(b) 3969
(c) 7371
(d) 1536
Ex17*, Find the prime factorization of
(a) 899
(b) 1147.
(e) 307
(f) 323