# Mathematical Tidbits

## Divisibility Tests

A positive integer is divisible by the given integer if the condition is met.

1. Every positive integer is divisible by 1.
2. If the last digit is a 0, 2, 4, 6, or 8
3. If the sum of the digits is divisible by 3
4. If the last two digits are divisible by 4
5. If the last digit is a 0 or 5
6. If the number is divisible by both 2 and 3.
7. Check it on the calculator
8. If the last three digits are divisible by 8
9. If the sum of the digits is divisible by 9
10. If the last digit is a 0
11. Subtract the sum of the digits in the even positions (2nd digit, 4th digit, etc) from the sum of the digits in the odd positions (1st digit, 3rd digit, etc). If this difference is divisible by 11, then the number is divisible by 11.

A test may be constructed for numbers such as 12, 15, and 18 according to the following rule. If a number can be factored so that the factors are relatively prime (that is, they have no common factors besides one), then the test for divisibility for that number the requirement that the number be divisible by the factors. 12 is a factor if 3 and 4 are both factors, but not necessarily if 2 and 6 are factors. 18 is a factor if 2 and 9 both are, but not necessarily if 3 and 6 both are. 14 is a factor if both 2 and 7 are, but there isn't an easy test for 7.

## Properties of Real Numbers

Closure
The set of real numbers is closed under addition and multiplication. This means that the sum of two real numbers is a real number and the product of two real numbers is a real number. The set of real numbers is also closed under subtraction (the difference of two real numbers is a real number), but not under division (the quotient of two real numbers may not be a real number - ie, division by zero does not yield a real number)
Commutative
The set of real numbers is commutative under addition and multiplication. This means that the order of the terms (addition) or factors (multiplication) is irrelevant to the answer. $$a + b = b + a$$ and $$a b = b a$$. The set of real numbers is not commutative with respect to subtraction and division, however: $$a - b \neq b - a$$ and $$\frac {a}{b} \neq \frac {b}{a}$$.
Associative
The set of real numbers is associative under addition and subtraction. This means that the grouping of terms (addition) or factors (multiplication) is irrelevant to the answer. $$(a+b)+c = a+(b+c)$$ or $$(ab)c = a(bc)$$. The set of real numbers is not associative with respect to subtraction and division, however: $$(a-b)-c \neq a-(b-c)$$ and $$(a/b)/c \neq a/(b/c)$$.
Identity
There is an additive identity and a multiplicative identity. The identity is the number that you can add or multiply by and get the same answer you started with. The additive identity is zero (0) and the multiplicative identity is one (1). Subtraction and division are defined in terms of addition and multiplication and the same identities hold.
Inverse
There is an additive inverse for all real numbers, and a multiplicative inverse all real numbers except for the additive identity zero (0). The sum of a number and its additive inverse is the additive identity zero (0). Another name for additive inverse is opposite. The product of a number and its mulitplicative inverse is the multiplicative identity one (1). Another name for multiplicative inverse is reciprocal. Every number except zero (0) has a reciprocal.
Distributive
There isn't a separate distributive property for addition and multiplication like there were with the other five properties. This is because the distributive property combines addition and multiplication. Stated simply, it says that "Multiplication distributes over addition". The left distributive property is: $$a(b+c)=ab + ac$$, and the right distributive property is $$(a+b)c = ac + bc$$. With real numbers, it is not important to distinguish between the left and right distributive properties because of commutativity. When we talk about Matrices, which aren't commutative under multiplication, then we must distinguish between the left and right properties

## Fundamental Theorem of Arithmetic

Every integer greater than one is either a prime number number or can be expressed as an unique product of prime numbers.

Prime factorization is the technique used to find that unique factorization.

Primes appear many times in arithmetic, hence the reason this is the fundamental theorem.

• The Least Common Multiple (LCM) is found by taking each prime factor in all the terms the most number of times it appears in any of the terms.
• The Greatest Common Factor (GCF) is found by taking each prime factor in all the terms the least number of times it appears in any of the terms.
• When reducing fractions, it is easy to see what to reduce if the numerator and denominator have been written in terms of primes.
• Prime numbers can be used in the divisibility tests to arrive at divisibility tests for composite numbers.
• The index of a radical must be larger than the exponent on each prime factor in the radicand for a radical to be in simplified radical form.

## Fundamental Theorem of Algebra

Every polynomial in one variable of degree n > 0 with real coefficients has at least one complex solution

A corollary which almost always follows is...

Every polynomial in one variable of degree n > 0 with real coefficients has exactly n complex solutions (not necessarily unique)

## First Fundamental Theorem of Calculus

If f is continuous on the closed interval [a,b] and F is an antiderivative of f on [a,b], then the definite integral from x=a to x=b of f(x) with respect to x is the difference between F(b) and F(a). $\int_a^b f(x)\, dx = F(b) - F(a)$

## Second Fundamental Theorem of Calculus

If f is continuous on an open interval containing x=a, then for every x in the interval, the derivative of the antiderivative is the original function. $\frac{d}{dx} \left [ \int_a^x f(t)\, dt \right ] = f(x)$

## Fundamental Theorem of Linear Programming

If a linear programming problem has a solution, then it will occur at one or more corner points or on the boundary between two corner points.

## Fundamental Theorem of Linear Programming - Duality

If f is objective function of a linear programming maximization problem and g is the objective function from the corresponding dual problem, then the maximization problem for f has a solution if and only if the minimization problem for g has a solution.