Notes on Integers

Parity of Integer

Sum & Subtraction

• Like terms always results in an even integer.
  • E + E = E
  • E - E = E
  • O + O = E
  • O - O = E
• Unlike terms results in an odd integer.
  • E + O = O
  • E - O = O

Product

• As long as there is 1 even factor in a product of two factors, the product is always even.
  • E * E = E
  • E * O = E
• Both factor terms in a product must be odd for an odd factor.
  • O * O = O
• Any integer multiplied to even and the result multipled to any integer is always even.
  • X * E or O = E
• 2 mulitplied by whether odd or even integer always gives an even integer.
  • 2 * E = E
  • 2 * O = E

Expression

• Use logical reasoning to deduce conditions of parity for individual terms in an expression.
• Use 4 case testing (use simple numbers 1, 2) method pluggging simple integers into an expression.
  1. Both P & Q are even.
  2. Both P & Q are odd.
  3. P is odd, Q is even.
  4. Q is odd, P is even.

Factors

Basics

• Factors of 12 are $$\set{1, 2, 3, 4, 6, 12}$$
• Multiples of 12 are $$\set{12, 24, 36, 72...}$$
• Visually:

        72
        |
        36          Multiple zone
        |
        24 
  ------|-------
        12
        /  \
       2    6       Factor zone
      / \    / \
     2  3  2  3 

• Theorem: Every integer greater than 1 can be represented as a product of prime numbers.
• Prime number is an integer greater than 1 whose factors are 1 and itself only.
• Prime number can be divided without remainder (i.e. evenly) by either 1 and itself only.
• 2 is the only even prime.
• All primes except 2 are odd.
• Common primes < 60: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59
• Equivalent statements:
  • r is factor of Q
  • r is divisior of Q
  • Q is divisible by r
  • Q is multiple of r
• Product, subtraction or sum of two multiples of an integer always gives its another multiple.

GCF and LCM

• GCF of A and B is the product of highest common prime factors present in both terms.
• If, $$ A = NY; B = NX $$ (where N is the GCF of A and B) then $$ LCM = X * Y * N $$
• $$ LCM=\frac{AB}{GCF} $$

Count of Factors

Let, $$ N = 2^5.3^2.7 $$ Here, $$\set{5,2,1}$$ are the exponents. Adding 1 to each exponent we get, $$\set{6,3,2}$$ Multiplying each new terms, we get, $$6.3.2=36$$ which is the total number of factors of N Multiplying each corresponding new term of odd prime factors (pf 3 and 7), we get $$3.2=6$$ which is the number of odd factors of N

Divisibility of Large Integers

• For 4, last 2 digits must be divisible by 4.
• For 3, sum of digits must be divisible by 3.
• For 9, sum of digits must be divisible by 9.
• For 5, last digit must be divisible by 5.
• For 6, must be divisible by both 2 and 3.

Integer-Quotient Equation

D = Dividend; S = Divisor; Q = Quotient; r = Remainder

• $$\frac{D}{S}=Q+\frac{r}{S}$$
• $$D=SQ+r$$