MathDB

HMMT Algebra/NT 2019/6: (a√2 + b√3) mod √2 + (a√2 + b√3) mod √3 = √2

Source:

February 17, 2019

HMMTalgebra

Problem Statement

For positive reals

p

and

q

, define the remainder when

p

and

q

as the smallest nonnegative real

r

such that

\tfrac{p-r}{q}

is an integer. For an ordered pair

(a, b)

of positive integers, let

r_1

and

r_2

be the remainder when

a\sqrt{2} + b\sqrt{3}

is divided by

\sqrt{2}

and

\sqrt{3}

respectively. Find the number of pairs

(a, b)

such that

a, b \le 20

and

r_1 + r_2 = \sqrt{2}