BASIC NUMBER THEORY
All natural numbers can be written in a special form. This form is uniquely
named “prime factorization”; it is indeed aptly named. To grasp this
rigorous concept, imagine any natural number n.
The “prime factorization” process must commence in this fashion:
1) Write the desired number.
2) Determine whether the number is divisible by prime p.
3) Divide n by p, writing the factors underneath the initial value,
indicating a distinct position for each.
4) Repeat this process with the next natural number acquired. Divide
this n_1 by p_1, meticulously indicating the position.
5) Once the number has been thoroughly “prime factorized”, there
should be a list of prime numbers at the bottom of each “branch”.
NOTE: Prime numbers are natural numbers that are divisible by itself and
one, excluding one! Memorize this! It’s important!
Prime factorization can be utilized in a myriad of ways, namely problems
that involve factors or basic number theory.
Let’s introduce a very simple example:
Suppose natural number n has prime factorization (p_a)^a X (p_b)^b …
(p_n)^n. (The dots assume an arbitrary length.)
IMPORTANT: Therefore, the number of factors of n is (a+1)(b+1)…(n+1).
EXAMPLE ONE: Determine the number of factors in 45.
45
5
5
9
3
3
The prime factorization of 45 = 5^1 X 3^2.
Looking at our formula, 45 would have (1+1)(2+1) = 6 factors.
EXAMPLE TWO: Determine the number of factors in 256.
Try using the “prime factorization” technique for yourself this time! It’s also
called the factor tree; it’s not difficult to see why.
The prime factorization of 256 = 2^8.
Once again applying our formula, 256 would have (8+1) = 9 factors.
IMPORTANT: Note that all perfect squares have an odd number of factors.
Challenge Problem 1: Using the “number-of-factors” theorem, prove that
any perfect square n^2 ( for n>1) must have an odd number of factors.
Prime factorization is useful in many areas of number theory. In fact, it
forms the fundamentals of the field; to some, there is nothing more
aesthetically pleasing or intellectually satisfying than studying primes.
Lowest Common Multiple (LCM) = the number that first appears on the list
of the multiples of each of both numbers. Technically speaking, it is the
number with the prime factorization of the largest exponent of each
individual prime factor of both numbers.
Greatest Common Factor (GCF) = the number that last appears on the list of
factors of each of both numbers from least to greatest. In mathematical
jargon, it is the number with the prime factorization of the smallest
exponent of each individual prime factor of both numbers.
IMPORTANT: LCM (a,b) X GCF (a,b) = a X b
(The above formula comes in very handy sometimes when tackling
especially challenging problems.)
PROBLEM SETS:
1) What is the prime factorization of 61732?
2) How many factors does 444444 have?
3) The number 100 has (x^2-2x+1) number of factors. Find x.
4) p(p+2)(p+4)(p+6)(p+104168) is even, where p is a prime number. How
many possible values of p are there?
Challenge Problem Two: Prove that the “number-of-factors” theorem.
5) What is the LCM of numbers 72 and 56?
6) What is the GCF of numbers 612 and 10798?
7) a + b = 30. LCM (a, b) = 88. GCF (a, b) = 2. What is a^2 + b^2?
8) Natural numbers a and b are relatively prime if the GCF is 1. Name all
numbers relatively prime between 1 and 50 that are relatively prime to 2.
9) What is the greatest prime divisor in: 1+2+3+… +70?
10) LCM (40,72) = a
GCF (176, 1021) = b
Number of factors of 1024 x 343 x 125 x 729 x 56 = c
To the nearest whole number, find (a^2 + b^2 + c^2) / 10000.