← Public decks·Science

Competition Mathematics Flashcards

High-school / contest math flashcards — number theory, GCD of exponential forms, and prime factor reasoning.

16 cards· by GuruOwl

Make a deck like this from your own PDF — free.

Try it
  1. 01
    Find the greatest common divisor of 2^1001 − 1 and 2^1012 − 1.
    2047 (Because gcd(2^a − 1, 2^b − 1) = 2^gcd(a,b) − 1, and gcd(1001, 1012) = 11, so 2^11 − 1 = 2047.)
    Identity: gcd(2^m − 1, 2^n − 1) = 2^gcd(m,n) − 1.
    number-theorygcdgsc-priority
  2. 02
    State the identity for gcd(2^m − 1, 2^n − 1).
    gcd(2^m − 1, 2^n − 1) = 2^gcd(m, n) − 1
    number-theorygcd
  3. 03
    A number x equals 2^15 · 3^6. What number’s cube equals x?
    288 (Need y^3 = 2^15 · 3^6 ⇒ y = 2^5 · 3^2 = 32 · 9 = 288.)
    exponentsgsc-priority
  4. 04
    Suppose that a, b, and c are positive integers satisfying a relationship that evaluates to 150; find a + b + c (contest-style factor problem). What general method do you use?
    Factor the equation, enumerate positive-integer factor triples (a,b,c) consistent with constraints, then compute a+b+c for the unique (or intended) solution.
    GSC query truncated the exact equation; use prime-factor / divisor enumeration once the full Diophantine form is known.
    contestgsc-priority
  5. 05
    If n = 2^a · 3^b, when is n a perfect cube?
    When every exponent is a multiple of 3 (a ≡ 0 mod 3 and b ≡ 0 mod 3).
    exponents
  6. 06
    If n = 2^a · 3^b, when is n a perfect square?
    When a and b are both even.
    exponents
  7. 07
    Compute gcd(2^12 − 1, 2^18 − 1).
    2^6 − 1 = 63 (gcd(12,18)=6.)
    number-theorygcd
  8. 08
    What is 2^10 − 1?
    1023
    arithmetic
  9. 09
    What is 2^11 − 1?
    2047
    arithmetic
  10. 10
    Factor 2^6 − 1.
    63 = 7 × 9 = 3^2 × 7 (Also 2^6 − 1 = (2^3 − 1)(2^3 + 1) = 7 × 9.)
    factoring
  11. 11
    Why does gcd(a,b) = gcd(b, a mod b)?
    Any common divisor of a and b also divides any integer linear combination, including a − qb; so common divisors of (a,b) equal those of (b, a mod b). This is the Euclidean algorithm.
    number-theory
  12. 12
    Use Euclid’s algorithm: gcd(1001, 1012).
    1012 − 1001 = 11, and 1001 = 91 × 11, so gcd = 11.
    number-theorygcd
  13. 13
    If x = 2^15 · 3^6, what is the cube root of x in prime factorization form?
    2^5 · 3^2
    exponents
  14. 14
    If x = 2^15 · 3^6, what is √x in prime factorization form?
    Not an integer power of primes with integer exponents for a perfect square root — 15 is odd, so x is not a perfect square. (√x = 2^{15/2} · 3^3.)
    exponents
  15. 15
    LCM vs GCD for two positive integers a and b: what identity links them?
    lcm(a,b) · gcd(a,b) = a · b
    number-theory
  16. 16
    Is 2047 prime?
    No. 2047 = 23 × 89 (= 2^11 − 1, a composite Mersenne number).
    primes