Discrete Math: Prove or disprove that among ANY 501 numbers are chosen from the

ID: 3012353 • Letter: D

Question

Discrete Math:
Prove or disprove that among ANY 501 numbers are chosen from the numbers 1, 2, …, 1000 there must be at least two numbers such that one divides the other.
Discrete Math:
Prove or disprove that among ANY 501 numbers are chosen from the numbers 1, 2, …, 1000 there must be at least two numbers such that one divides the other.
Discrete Math:
Prove or disprove that among ANY 501 numbers are chosen from the numbers 1, 2, …, 1000 there must be at least two numbers such that one divides the other.
Discrete Math:
Prove or disprove that among ANY 501 numbers are chosen from the numbers 1, 2, …, 1000 there must be at least two numbers such that one divides the other.

Explanation / Answer

This can be proved by considering the number of primes upto 1000

We use the prime number theorem (an asymptotic approximation) for that

As per Prime number theorem

pi(n) = n/ln (n)

i.e. the number of primes upto n is given by n upon natural log of n

so for upto 1000

pi(1000) = 1000/ln 1000 =~ 144 primes

since, any 501 numbers choosen between 1 to 1000 can contain a maximum of 144 primes, therefore we will have a minimum of 357 composite no.s which are multiples of those primes.

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