Statement form. 'Prove that for all positive integers n, ___ is divisible by ___.'
To prove divisibility by d, write the expression as d⋅(integer).
Worked example. Prove that the sum of any three consecutive integers is divisible by 3.
Let the three consecutive integers be n, n+1, n+2.
Sum: n+(n+1)+(n+2)=3n+3=3(n+1).
Since n+1 is an integer, the sum is divisible by 3. ✓
Worked example. Prove that the sum of any five consecutive integers is divisible by 5.
Let n−2,n−1,n,n+1,n+2. (Centring at n is a clean choice.)
Sum: 5n. Divisible by 5. ✓
(Centring around n is a common trick — the symmetric terms cancel.)
Worked example. Prove that n3−n is divisible by 6 for every positive integer n.
Factorise: n3−n=n(n2−1)=n(n−1)(n+1)=(n−1)n(n+1).
This is a PRODUCT OF THREE CONSECUTIVE INTEGERS.
- Among any three consecutive integers, at least one is even (divisible by 2).
- Among any three consecutive integers, exactly one is divisible by 3.
So the product is divisible by both 2 and 3, hence by 6 (since gcd(2,3)=1). ✓
This 'invoke a property of consecutive integers' is a common AA SL Paper 1 technique.
Worked example (algebraic identity). Show that (2n+1)2−(2n−1)2=8n.
LHS =4n2+4n+1−(4n2−4n+1)=8n. ✓
This proves the difference of squares of consecutive odd numbers is always divisible by 8 — a classic Paper 1 result.