MathDB

The equation of line

\ell_1

24x-7y = 319

and the equation of line

\ell_2

12x-5y = 125

. Let

a

be the number of positive integer values

n

less than

2023

such that for both

\ell_1

and

\ell_2

there exists a lattice point on that line that is a distance of

n

from the point

(20,23)

. Determine

a

.
Proposed by Christopher Cheng
Solution.

\boxed{6}

Note that

(20,23)

is the intersection of the lines

\ell_1

and

\ell_2

. Thus, we only care about lattice points on the the two lines that are an integer distance away from

(20,23)

. Notice that

7

and

24

are part of the Pythagorean triple

(7,24,25)

and

5

and

12

are part of the Pythagorean triple

(5,12,13)

. Thus, points on

\ell_1

only satisfy the conditions when

n

is divisible by

25

and points on

\ell_2

only satisfy the conditions when

n

is divisible by

13

. Therefore,

a

is just the number of positive integers less than

2023

that are divisible by both

25

and

13

. The LCM of

25

and

13

325

, so the answer is

\boxed{6}

Problem Statement