Let us tackle comprehensive problems that combine the properties of integers studied so far.
Example ① (Combining congruences and factorization)
Prove that n³+2n is a multiple of 3.
n³+2n = n(n²+2) = n(n²−1+3) = n(n−1)(n+1)+3n
The product n(n−1)(n+1) is a product of three consecutive integers → a multiple of 3; 3n is also a multiple of 3 → the whole expression is a multiple of 3 ■
（Alternative solution）For each case n≡0,1,2(mod 3), n³+2n≡0+0=0、1+2=3≡0、8+4=12≡0(mod 3) ■
Prove that n³+2n is a multiple of 3.
n³+2n = n(n²+2) = n(n²−1+3) = n(n−1)(n+1)+3n
The product n(n−1)(n+1) is a product of three consecutive integers → a multiple of 3; 3n is also a multiple of 3 → the whole expression is a multiple of 3 ■
（Alternative solution）For each case n≡0,1,2(mod 3), n³+2n≡0+0=0、1+2=3≡0、8+4=12≡0(mod 3) ■
Example ② (Pythagorean triples)
Find all positive integer triples (a,b,c) satisfying a² + b² = c² when c=5.
a²+b²=25。Assume a≤b. For a=1:1+b²=25→b²=24（×）. For a=2:4+b²=25→b²=21（×）. For a=3:9+16=25→b=4(✓).
Answer:(a,b,c) = (3,4,5) and (4,3,5)
Find all positive integer triples (a,b,c) satisfying a² + b² = c² when c=5.
a²+b²=25。Assume a≤b. For a=1:1+b²=25→b²=24（×）. For a=2:4+b²=25→b²=21（×）. For a=3:9+16=25→b=4(✓).
Answer:(a,b,c) = (3,4,5) and (4,3,5)
Example ③ (Inequalities and integers)
Find all positive integer pairs (x,y) satisfying 1/x + 1/y = 1/3（x≤y）.
3(x+y)=xy → xy−3x−3y=0 → (x−3)(y−3)=9
Since x≤y, x−3≤y−3. 9=1×9=3×3（pairs of positive divisors）
(x−3,y−3)=(1,9)→(x,y)=(4,12). (x−3,y−3)=(3,3)→(x,y)=(6,6)
Answer:(x,y) = (4,12),(6,6)
Find all positive integer pairs (x,y) satisfying 1/x + 1/y = 1/3（x≤y）.
3(x+y)=xy → xy−3x−3y=0 → (x−3)(y−3)=9
Since x≤y, x−3≤y−3. 9=1×9=3×3（pairs of positive divisors）
(x−3,y−3)=(1,9)→(x,y)=(4,12). (x−3,y−3)=(3,3)→(x,y)=(6,6)
Answer:(x,y) = (4,12),(6,6)
Strategies for comprehensive problems
- “Proof of a multiple” → product of consecutive integers or casework using mod
- “Integer solutions of an equation” → transform into a product and list the factors
- “1/x+1/y=1/k type” → transform into (x−k)(y−k)=k²
Practice Problems
- When n is an integer, prove that n²+n is always even.
- Find all positive integer pairs (x,y) satisfying 1/x + 1/y = 1/4（x≤y）.
- Find all positive integers satisfying a²+b²=c² when c=13（a≤b）.
Answers and Explanations
- n²+n=n(n+1)。The product of two consecutive integers is always even (one of them is even) → even ■
- (x−4)(y−4)=16。Pairs of positive divisors: (1,16),(2,8),(4,4) → (x,y)=(5,20),(6,12),(8,8)。Answer: (5,20),(6,12),(8,8)
- a²+b²=169。a=5:25+b²=169→b²=144→b=12(✓)。a=12:144+25=169→(12,5)（excluded because a≤b）。
(a,b,c)=(5,12,13)