Properties of Integers ② Summary
- Congruence: a≡b(mod m) ⟺ m|(a−b)
- Operations with congruences: addition, subtraction, multiplication, and exponentiation are possible
- Periodicity: remainders of powers are periodic
- Base-n conversion: base n→base 10 (expansion), base 10→base n (division remainders)
- Integer problems: narrow down possibilities using products, parity, and mod
Comprehensive Practice Problems
- Find the remainder when 250 is divided by 7.
- Find n such that (123)n = 38 (n is a natural number).
- Find all integer solutions satisfying xy + x − y = 7.
Answers and Explanations
- 2³=8≡1(mod 7). 50=3×16+2 → 250=(2³)16×2²≡116×4=4(mod 7) → remainder4
- n²+2n+3=38 → n²+2n−35=0 → (n+7)(n−5)=0 → n=5（n>0）. Check: (123)5=1×25+2×5+3=38 ✓
- (x+1)(y−1)=6（rearranging after adding +1 to both sides: xy+x−y+1=8→(x+1)(y−1)=8→incorrect. xy+x−y−1=6→x(y+1)−(y+1)=6→(x−1)(y+1)=8）.
Pairs: (x−1,y+1)=(1,8),(2,4),(4,2),(8,1),(−1,−8),(−2,−4),(−4,−2),(−8,−1)
(x,y)=(2,7),(3,3),(5,1),(9,0),(0,−9),(−1,−5),(−3,−3),(−7,−2)