What are the factors of 36? If you remember any high school math then that question probably was not that hard (the answer is 1, 2, 3, 4, 6, 9, 12, 18, and 36). Just in case though, let’s go over exactly what a factor is.
Factoring, or factorization, is the process by which a mathematical expression is rewritten as the product of simpler mathematical expressions. The elements that multiply together to equal the more complex expression are called that expression’s factors. In other words, the factors of some number N are the numbers that you can multiply together to equal N.
“Mathematics is not about numbers, equations, computations, or algorithms: it is about understanding.” — William Paul Thurston
So with our original question, the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36 because 1 × 36 = 36, 2 × 18 = 36, 3 × 12 = 36, 4 × 9 = 36, and 6 × 6 = 36. Similarly, the factors of 16 are 1, 2, 4, 8, and 16; and the factors of 21 are 1, 3, 7, and 21. Some numbers, like 23, are unique in that they only have 2 factors, 1 and themselves. These numbers are called prime numbers and they have some very special properties.
Numbers are not the only things that can have factors. Algebraic expressions, like x2−5x+6, have factors as well. Specifically, the factors of x2−5x+6 are simpler expressions (x−2) and (x−3). One can multiply (x−2) by (x−3) to get x2−5x+6 and so they count as its factors.
Mathematical Properties Of Factors
Immediately, one can start to see some patterns that arise with factors. For example, every even number has 2 as a factor. Consequently, no even number can be prime, as they always have more factors than just one and themselves (except for the number 2 itself, which is notable for being the sole even prime number).
For all numbers N, the largest factor, aside from N itself, is less than or equal to half of N. This is because there is no number greater than half of N that you can multiply to equal N. Also, notice that all numbers can be written as the product of some set of prime numbers. For example, 27 can be expressed 27= 3×9 = 3×32 = 3×3×3. The expression 3×3×3 is the prime factorization of 27. The existence of prime factorizations is a consequence of the fundamental theorem of arithmetic, the fact that every number greater than 1 is either prime itself, or is the product of some prime numbers.
“Math is the language of the universe. So the more equations you know, the more you can converse with the cosmos.” — Neil DeGrasse Tyson
Not every expression can be factored, though. Expressions that cannot be factored are called irreducible polynomials and are said to be in their simplest form. Irreducible polynomials cannot be written as the product and any simpler elements. Whether or not polynomial is irreducible is relative to a specific domain. x2−2 = (x−√2)(x+√2) is irreducible over the domain of the integers, as it cannot be written as the product of any 2 integers (√2 is not an integer). x2−2 = (x−√2)(x+√2) however, is reducible with respect to the real numbers, as √2 is a real number. Some polynomials are absolutely irreducible; that is irreducible across every domain of numbers. For example, the polynomial xn+yn−1 (sometimes called the Fermat curve) is irreducible for any positive number n. This means that it is impossible to express xn+yn−1 as the product of any two mathematical objects.
Applications Of Factoring
Factoring has a number of uses in calculus, physics, algebra, trig, and more complex mathematics. Primarily, factoring allows one to simplify a complex expression and rewrite as the product of simpler parts. In doing so, often one can highlight mathematical relationships otherwise hidden.
Here is an example: say we have the messy expression 4√162. To simplify this we can factor the expression and get 4√(3×3×3×3×2) = 4√(34×2) = 34√2. 34√2 is much easier to deal with than 4√162, and so factoring has saved us a huge headache.
Alternatively, factoring can be useful in situations where one is attempting to model real-world phenomena with mathematics. Let’s say I am tossing bowling balls off of the Leaning Tower of Pisa (Galileo would be proud). The motion of each bowling ball with respect to time can be represented by the equation ƒ(x)=−x2+x+6.
“Pure mathematicians just love to try unsolved problems – they love a challenge.” — Andrew Wiles
Suppose we want to know how long it takes for each bowling ball to hit the ground. One way to figure this out is to find the roots of the equation; the x values for which the equation equals zero. Naturally, we would interpret the roots of the equation as representing the point where the bowling ball hits the ground. We can figure out the roots of the equation by factoring: ƒ(x)=−x2+x+6 = -(x+2)(x−3). When multiplying, if one of the factors equals zero, then the whole expression equals zero. Therefore, the roots of the equation are -2 and 3, respectively, as each term of -(x+2)(x−3) is equal to zero for those x values. Of course, a value of -2 seconds would not make sense in our situation. So the answer to our original question is 3: the bowling ball hits the ground after 3 seconds By factoring the equation of motion for the bowling ball, we were able to determine how long it would take for each bowling ball to hit the ground.
Additionally, factoring, or more specifically the difficulty of factoring extremely large numbers, plays a role in cryptography. Essentially, encrypted messages contain both “public key” and a “secret key.” The public key consists of a number that is the product of two large prime numbers and is used to initially encrypt the message. The secret key consists of those two prime factors and is necessary to decrypt the message. Extremely large numbers are very difficult to factor into primes, so unless one already knows the secret key, it is practically impossible to break the encryption given the current state of computer technology. In fact, given the current state of computing technology, if one tried to brute force break a common 128-bit encryption, it would take about 266 seconds to test all possible combinations. To be clear, that is equivalent to about 2, 158 000, 000, 000 years—150 times the current age of the universe (~13 billion years). Recent advances in quantum computation could, however, change the state of current encryption practices.
Factoring is one of the most basic and useful mathematical operations. In fact, recent studies show that relatively simple mathematical techniques, like factoring, are more useful to aiding mathematical understanding than more complex mathematical concepts. Factoring has useful applications all across the industry, so next time you are in math class make sure to pay attention.