Question # There are 60 people taking the Discrete Mathematics classthey have a distinct roll numbers from 1 to 60. Now if they pick arbitrarily 31 students. Claim: there exists two students with roll number a and b such that b is divisible by a.

Discrete math
ANSWERED There are 60 people taking the Discrete Mathematics classthey have a distinct roll numbers from 1 to 60. Now if they pick arbitrarily 31 students.
Claim: there exists two students with roll number a and b such that b is divisible by a. 2021-08-06

Step 1
Let $$\displaystyle{X}={\left\lbrace{1},\ {2},\ {3},\cdots,{60}\right\rbrace}$$ with $$\displaystyle{\left|{X}\right|}={60}$$ and
Let $$\displaystyle\phi={\left\lbrace{1},\ {7},\ {3},\cdots,{40}\right\rbrace}$$ with $$\displaystyle{\left|\phi\right|}={31}$$
Claim: $$\displaystyle\exists\ {a},\ {b}\in\ {X}$$ such that $$\displaystyle{\frac{{{b}}}{{{a}}}}={k}\in\ {X}$$
Suppose not then $$\displaystyle{a},\ {b}\in\ {X}$$
but if $$\displaystyle{b}={3}$$ and $$\displaystyle{a}={2}$$
with $$\displaystyle{a}{<}{b}$$ is not satisfying the condition $$\displaystyle{\frac{{{b}}}{{{a}}}}={k}\in\ {X}$$
also when $$\displaystyle{a}{>}{b},$$ with $$\displaystyle{a}={3}$$ and $$\displaystyle{b}={2}$$ still it
is not satisflying the condition $$\displaystyle{\frac{{{b}}}{{{a}}}}={k}\in\ {X}$$
it satisflying the condition $$\displaystyle{\frac{{{b}}}{{{a}}}}={k}\in\ {X}$$
only if $$\displaystyle{a}={b}$$
therefore, $$\displaystyle\exists{a},\ {b}\in{X}$$ such that $$\displaystyle{\frac{{{b}}}{{{a}}}}={k}\in\ {X}$$, only if $$\displaystyle{a}={b}$$