Properties of Integers ① Summary
- Prime factorization: n=paqb… → number of divisors=(a+1)(b+1)…
- Sum of divisors: (1+p+…+pa)(1+q+…+qb)…
- GCD×LCM=a×b
- Euclidean algorithm: GCD(a,b)=GCD(b,a mod b)
- Diophantine equation: find a particular solution using the Euclidean algorithm → general solutions form an arithmetic progression
Comprehensive Practice Problems
- Find the number and sum of the divisors of 504.
- When GCD(a,b)=8 and LCM(a,b)=240, find the value of a×b and all possible pairs (a,b) (a≤b).
- Find the general integer solution of 11x + 8y = 5, and find y when x is the smallest positive integer.
Answers and Explanations
- 504=2³×3²×7. Number of divisors=(3+1)(2+1)(1+1)=24. Sum=(1+2+4+8)(1+3+9)(1+7)=15×13×8=1560
- a×b=8×240=1920. a=8m, b=8n（m
- 11×3+8×(−4)=1 → 11×15+8×(−20)=5 → particular solution (15,−20). General solution: x=15+8t, y=−20−11t. x>0 is smallest when t=−1, giving x=7; t=−2 gives x=−1(NG) → when the smallest positive integer isx=7, y=−20−11×(−1)=−9