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

Free flashcards from Peabody at peabody.kids/flashcards/math-number-theory