3

Part of 2021 Czech-Polish-Slovak Junior Match

Problems(2)

max no of 6 pieces crosses in a 6x11 sheet of paper

Source: Czech-Polish-Slovak Match Junior 2021, individual p3 CPSJ

A cross is the figure composed of

6

unit squares shown below (and any figure made of it by rotation). https://cdn.artofproblemsolving.com/attachments/6/0/6d4e0579d2e4c4fa67fd1219837576189ec9cb.png Find the greatest number of crosses that can be cut from a

6 \times 11

divided sheet of paper into unit squares (in such a way that each cross consists of six such squares).

combinatoricscombinatorial geometry

gcd (a,b)=50! , lcm(a,b)=(50!)^2

Source: 2021 Czech-Polish-Slovak Match Junior, team p3 CPSJ

Find the number of pairs

(a, b)

of positive integers with the property that the greatest common divisor of

a

b

1\cdot 2 \cdot 3\cdot ... \cdot50

, and the least common multiple of

a

b

1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2

number theorygreatest common divisorleast common multipleLCMGCD