1

Part of 1983 Vietnam National Olympiad

Problems(2)

Easy number theory: 2^b-1 divides 2^a+1

Source: Vietnam MO 1983 P1

Are there positive integers

a, b

b \ge 2

2^a + 1

is divisible by

2^b - 1

modular arithmeticnumber theorygreatest common divisornumber theory proposed

Expressing 1 as sum of 6 and 9 reciprocals of odd integers.

Source: Vietnam MO 1983 P4

Show that it is possible to express

1

6

, and as a sum of

9

reciprocals of odd positive integers. Generalize the problem.

number theory unsolvednumber theory