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9
Other Types of Number
Prime Numbers: These are numbers that have two factors only, one of which
is 1, and the second of which is the number itself. All primes are odd, except
for 2. The first ten primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. 1 is not
prime since it only has one factor, which is itself.
Square Numbers: These are numbers derived by multiplying a positive integer
by itself. For example, 4 * 4 = 16, so 16 is a square. 4 itself is a square, as it is
the product of 2 and itself.
Cube Numbers: These are numbers derived by multiplying a positive integer
by itself, twice. For example, 3 * 3 * 3 = 27, so 27 is a cube.
Reciprocals: The reciprocal of any number is the result of dividing 1 by that
number. For example, the reciprocal of 2 is 1/2. For a real number, n, the
reciprocal is 1/n.
Using Prime Factors
Prime factors: These are prime numbers that, when multiplied together, give a
product of a composite (non-prime) number. For example, the prime factors of
6 are 2 and 3. The prime factors of 12 are 2, 2, and 3.
To find the prime factors of any composite number, a factor tree can be used.
This involves splitting up a number into factors, and splitting up each of those
factors until all of them lead to primes. These cannot be split into anything but
1 and themselves.
If asked to write prime factors of a number with powers, the following example
may be used: 72 = (2*2*2) * (3*3).
If all the unique prime factors of a number appear an even number of times,
the number is a square. For example, (2*2*2*2) * (3*3) = 144, which is the
square of 12. If all the prime factors appear a number of times which is any
multiple of 3, the number is a cube. For example, (5*5*5) * (2*2*2*2*2*2) =
8000, which is the cube of 20.
Highest Common Factor: To find common factors, all the factors of two or
more numbers can be listed down. Factors appearing in all these lists are
common. The highest of these is known as the HCF, or highest common
factor. This is sometimes referred to as the GCD, or greatest common
denominator, as well. To find the HCF more efficiently, a number can be
decomposed into its prime factors, and then the common prime factors can be
multiplied to find the HCF.
Lowest Common Multiple: To find common multiples, the first few multiples of
two or more numbers can be listed down. Multiples appearing in all lists are