What is prime factorization?
Many of us are most familiar with prime factorization, which is a way of taking a num-
ber and breaking it down into its constituent primes. An example of prime factorization
is as follows: One finds the “simplest” representation of the given quantity in terms of
smaller parts—in the case of 15, the factors would be 1, 3, 5, and 15 (essentially, all the
numbers that will divide integrally into 15). Not that prime factorization is always that
easy. Larger numbers make it more difficult to factor, and many sophisticated prime
algorithms have been devised for larger—and different types—of numbers.What does the greatest common factormean?
The greatest common factor(or GCF; sometimes called highest common factor) of
two whole numbers is the largest whole number that is a factor of both. Take, for
example, the numbers 12 and 15: The factors of 12 are 1, 2, 3, 4, 6, and 12; the factors
of 15 are 1, 3, 5, and 15. Therefore, the common factors—or numbers in both lists of
factors—are 1 and 3; and the greatest (highest) common factor in this case is 3.
There is another method used to discover the GCF: listing the numbers’ prime
factors, then multiplying those numbers. For example, the prime factorizations of 12
and 15 are: 2 2 3 12 and 3 5 15. Notice that the prime numbers have 3 in
common; thus, the GCF is 3.
An example with larger numbers is to find the GCF of 36 and 54. Working it out by
the first method, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36; the factors of 54 are
1, 2, 3, 6, 9, 18, 27, and 54. The greatest (or highest) common factor of both numbers is- To work it out using prime factorization, the prime factorization of 36 is 2 2 3
3; the prime factorization of 54 is 2 3 3 3. Both these factorizations have one 2
and two 3s in common; thus, we multiply those common numbers, or 2 3 3 18.
FRACTIONS
What are properand improper fractions?
The word “fraction” to most of us means a part of something; in mathematics, it rep-
resents a type of numeral, in most cases the quotient of two integers, with the top
number called the numerator(the number of parts) and the bottom number the
denominator(how many parts the whole is divided into). Written out, the numerator
and denominator are separated by a “/” or “—”. A fraction is usually denoted by a/b, in
which “a” and “b” are whole numbers and “b” is not equal to zero. (For the explana-
tion of why you cannot divide by zero, see above).
A rational number between 0 and 1 can be represented by fractions (by the divi-
sion of two numbers). If the quotient is less than one, such as 1/2 or 2/5, then it is
98 called a proper fraction;if the quotient is greater than one—or, in other words, if the