MathDB

For any two coprime positive integers

p, q

, define

f(i)

to be the remainder of

p\cdot i

divided by

q

for

i = 1, 2,\ldots,q -1

. The number

i

is called a large number (resp. small number) when

f(i)

is the maximum (resp. the minimum) among the numbers

f(1), f(2),\ldots,f(i)

. Note that

1

is both large and small. Let

a, b

be two fixed positive integers. Given that there are exactly

a

large numbers and

b

small numbers among

1, 2,\ldots , q - 1

, find the least possible number for

q

.
Proposed by usjl

Problem Statement