MathDB

1

Part of 1996 Irish Math Olympiad

Problems(2)

compute f(n)

Source: Ireland 1996

7/1/2009
For each positive integer n n, let f(n) f(n) denote the greatest common divisor of n!\plus{}1 and (n\plus{}1)!. Find, without proof, a formula for f(n) f(n).
modular arithmeticgreatest common divisornumber theoryrelatively primenumber theory proposed
Fibonacci

Source: Ireland 1996

7/1/2009
The Fibonacci sequence is defined by F_0\equal{}0, F_1\equal{}1 and F_{n\plus{}2}\equal{}F_n\plus{}F_{n\plus{}1} for n≥0 n \ge 0. Prove that: (a) (a) The statement "F_{n\plus{}k}\minus{}F_n is divisible by 10 10 for all n∈N" n \in \mathbb{N}" is true if k\equal{}60 but false for any positive integer k<60 k<60. (b) (b) The statement "F_{n\plus{}t}\minus{}F_n is divisible by 100 100 for all n∈N" n \in \mathbb{N}" is true if t\equal{}300 but false for any positive integer t<300 t<300.
number theory proposednumber theory