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
asked 2021-08-05
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.

Answers (1)

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 \(\displaystyle{a}={b}\)

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