MathDB

1

Part of 1997 Irish Math Olympiad

Problems(2)

easy exercise

Source: Ireland 1997

7/3/2009
Find all pairs of integers (x,y) (x,y) satisfying 1\plus{}1996x\plus{}1998y\equal{}xy.
number theory proposednumber theory
abundant integers

Source: Ireland 1997

7/3/2009
Given a positive integer n n, denote by σ(n) \sigma (n) the sum of all positive divisors of n n. We say that n n is abundant abundant if σ(n)>2n. \sigma (n)>2n. (For example, 12 12 is abundant since \sigma (12)\equal{}28>2 \cdot 12.) Let a,b a,b be positive integers and suppose that a a is abundant. Prove that ab ab is abundant.
number theory unsolvednumber theory