Integers¶

• Divisibility: No remainder
• Divisor = Factor
• Multiple
• Rules
  • 2: Number is even
  • 3: sum of digits is divisible by 3
  • 4: 2 trailing digits divisible by 4
  • 5: last digit is 0 or 5
  • 6: Number divisible by 2 and 3
  • 7:
  • 8:
  • 9: Sum of digits divisible by 9
  • 10: Last digit is 0
• Prime numbers
• +ve integer with only 2 divisors: 1 and itself
• 1 is neither prime nor composite
• 2 is only even prime number
• Prime factorization
• Any integer > 1 is either prime or can be expressed be expressed as product of prime numbers
• If \(n= p^a q^b r^c \cdots\) where \(p, q, r, \dots\) are prime factors of \(n\), then total number of positive divisors of \(n\) is \((a+1)(b+1)(c+1) \cdots\)
• Squares of integers
• called perfect squares
• Prime factorization will always have even no of each prime
• Will always have odd numbers of +ve divisors
• GCD/HCF
• Names
  • Greatest Common Divisor
  • Greatest Common Factor
  • Highest Common Factor
• Greatest +ve common divisor shared by 2/more numbers
• LCM
• Least common multiple
• Smallest positive integer that is a multiple of both numbers
• \(\text{HCF}(x, y) \times \text{LCM}(x, y) = x \times y\)
• Operations with odd/even integers
• Product of odd numbers is always odd
  • Add/sub
  • Odd +- odd = even
  • Odd +- even = odd
  • Even +- even = even
  • Mul
  • Odd x odd = odd
  • Odd x even = even
  • even x even = even
  • Div
  • Even/Even can be anything
  • Odd/even = non-integer
  • Even/odd = non-integer or even integer
  • Odd/odd: non-integer or odd integer
• Consecutive integers
• Every \(n\)th number is divisible by \(n\)
• \(n\) consecutive integers \(\implies\) 1 number must be divisible by \(n\)
• Remainders
• Remainder \(\in\) [0, Divisor)
• Dividend = divisor x quotient + remainder
• If \(n/D = Q \text{ with } R\), then possible values of \(n\) are \(R + aD\), where \(a \ge 0\)

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