{
 "source": "https://peabody.kids/flashcards/math-number-theory",
 "id": "math-number-theory",
 "title": "Number Theory and Divisibility",
 "category": "math",
 "subject": "Math",
 "levels": [
  "3-5",
  "6-8",
  "9-12"
 ],
 "description": "Divisibility rules, factors, multiples, GCF, LCM, primes and modular ideas.",
 "tags": [
  "divisibility",
  "factors",
  "number theory"
 ],
 "frontLabel": "Question",
 "backLabel": "Answer",
 "cards": [
  [
   "Divisible by 2 if",
   "The last digit is even (0, 2, 4, 6, 8)"
  ],
  [
   "Divisible by 3 if",
   "The sum of the digits is divisible by 3"
  ],
  [
   "Divisible by 4 if",
   "The last two digits form a number divisible by 4"
  ],
  [
   "Divisible by 5 if",
   "The last digit is 0 or 5"
  ],
  [
   "Divisible by 6 if",
   "It is divisible by both 2 and 3"
  ],
  [
   "Divisible by 8 if",
   "The last three digits form a number divisible by 8"
  ],
  [
   "Divisible by 9 if",
   "The sum of the digits is divisible by 9"
  ],
  [
   "Divisible by 10 if",
   "The last digit is 0"
  ],
  [
   "Divisible by 11 if",
   "The alternating sum of the digits is divisible by 11"
  ],
  [
   "Is 2,346 divisible by 3?",
   "Yes, digit sum 15"
  ],
  [
   "Is 4,125 divisible by 9?",
   "No, digit sum 12"
  ],
  [
   "Is 1,716 divisible by 4?",
   "Yes, 16 is divisible by 4"
  ],
  [
   "Is 121 divisible by 11?",
   "Yes: 1 − 2 + 1 = 0"
  ],
  [
   "Factors of 12",
   "1, 2, 3, 4, 6, 12"
  ],
  [
   "Factors of 24",
   "1, 2, 3, 4, 6, 8, 12, 24"
  ],
  [
   "Factors of 36",
   "1, 2, 3, 4, 6, 9, 12, 18, 36"
  ],
  [
   "Factors of 48",
   "1, 2, 3, 4, 6, 8, 12, 16, 24, 48"
  ],
  [
   "Factors of 60",
   "1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60"
  ],
  [
   "Factors of 100",
   "1, 2, 4, 5, 10, 20, 25, 50, 100"
  ],
  [
   "Prime factorization of 12",
   "2² × 3"
  ],
  [
   "Prime factorization of 36",
   "2² × 3²"
  ],
  [
   "Prime factorization of 60",
   "2² × 3 × 5"
  ],
  [
   "Prime factorization of 100",
   "2² × 5²"
  ],
  [
   "Prime factorization of 72",
   "2³ × 3²"
  ],
  [
   "Prime factorization of 84",
   "2² × 3 × 7"
  ],
  [
   "Prime factorization of 90",
   "2 × 3² × 5"
  ],
  [
   "Prime factorization of 120",
   "2³ × 3 × 5"
  ],
  [
   "Prime factorization of 144",
   "2⁴ × 3²"
  ],
  [
   "GCF of 12 and 18",
   "6"
  ],
  [
   "GCF of 24 and 36",
   "12"
  ],
  [
   "GCF of 15 and 25",
   "5"
  ],
  [
   "GCF of 48 and 72",
   "24"
  ],
  [
   "GCF of 17 and 31",
   "1 (both prime)"
  ],
  [
   "LCM of 4 and 6",
   "12"
  ],
  [
   "LCM of 6 and 8",
   "24"
  ],
  [
   "LCM of 3 and 5",
   "15"
  ],
  [
   "LCM of 9 and 12",
   "36"
  ],
  [
   "LCM of 10 and 15",
   "30"
  ],
  [
   "Relationship of GCF and LCM",
   "GCF(a, b) × LCM(a, b) = a × b"
  ],
  [
   "Coprime (relatively prime)",
   "Two numbers whose only common factor is 1"
  ],
  [
   "Euclidean algorithm",
   "Find the GCF by repeatedly replacing the larger number with the remainder"
  ],
  [
   "Sieve of Eratosthenes",
   "A method that crosses out multiples to find primes"
  ],
  [
   "Fundamental theorem of arithmetic",
   "Every integer above 1 has exactly one prime factorization"
  ],
  [
   "Largest prime factor idea",
   "Test divisors only up to the square root"
  ],
  [
   "Number of factors of a prime",
   "2"
  ],
  [
   "Is there a largest prime?",
   "No. Euclid proved there are infinitely many primes"
  ],
  [
   "Is 97 prime?",
   "Yes"
  ],
  [
   "Is 101 prime?",
   "Yes"
  ],
  [
   "Is 111 prime?",
   "No: 3 × 37"
  ],
  [
   "Is 143 prime?",
   "No: 11 × 13"
  ],
  [
   "Is 221 prime?",
   "No: 13 × 17"
  ],
  [
   "Remainder of 17 ÷ 5",
   "2"
  ],
  [
   "17 mod 5",
   "2"
  ],
  [
   "Modular arithmetic",
   "Arithmetic where numbers wrap around after reaching a set value"
  ],
  [
   "Clock arithmetic example",
   "5 hours after 10 o'clock is 3 o'clock (15 mod 12)"
  ],
  [
   "Even + even",
   "Even"
  ],
  [
   "Odd + odd",
   "Even"
  ],
  [
   "Odd + even",
   "Odd"
  ],
  [
   "Odd × odd",
   "Odd"
  ],
  [
   "Even × any integer",
   "Even"
  ],
  [
   "Sum of the first n odd numbers",
   "n²"
  ],
  [
   "Sum of the first n whole numbers",
   "n(n + 1)/2"
  ],
  [
   "Sum of 1 to 100",
   "5,050"
  ],
  [
   "Goldbach's conjecture",
   "Every even number above 2 is the sum of two primes (unproven)"
  ],
  [
   "Fermat's Last Theorem",
   "No positive integers satisfy aⁿ + bⁿ = cⁿ for n above 2; proved in 1994-95"
  ],
  [
   "Pythagorean triple",
   "Three whole numbers a, b, c with a² + b² = c²"
  ],
  [
   "Common Pythagorean triples",
   "3-4-5, 5-12-13, 8-15-17, 7-24-25"
  ]
 ]
}