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Question Number 205726 by hardmath last updated on 28/Mar/24
101 is chosen arbitrarily from the numbers 1,2,3,...,199,200. Prove that two of these selected numbers can be found such that one is divisible by the other.
$$ \\ $$101 is chosen arbitrarily from the numbers 1,2,3,…,199,200. Prove that two of these selected numbers can be found such that one is divisible by the other.
Answered by A5T last updated on 29/Mar/24
All n_i ∈{1,2,3,...,200} are of the form 2^k_i a_j where a_j ∈{1,3,5,...,199}, 2^k_i is the largest power of two that divides n_i . Since there exists only 100 possible different values for a_j . When there are 101 different numbers, there exists atleast two,say n_i_i and n_(i_2 ,) such that a_j_1 =a_j_2 . The larger of n_i_1 or n_i_2 must divide the other :(n_i_1 /n_i_2 )=((2^k_1 a_j_2 )/(2^k_2 a_j_1 ))=2^(k_1 −k_2 ) ;(n_i_2 /n_i_1 )=2^(k_2 −k_1 ) .
$${All}\:{n}_{{i}} \in\left\{\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{200}\right\}\:{are}\:{of}\:{the}\:{form}\:\mathrm{2}^{{k}_{{i}} } {a}_{{j}} \\ $$$${where}\:{a}_{{j}} \in\left\{\mathrm{1},\mathrm{3},\mathrm{5},…,\mathrm{199}\right\},\:\mathrm{2}^{{k}_{{i}} } \:{is}\:{the}\:{largest}\:{power} \\ $$$${of}\:{two}\:{that}\:{divides}\:{n}_{{i}} .\: \\ $$$${Since}\:{there}\:{exists}\:{only}\:\mathrm{100}\:{possible}\:{different} \\ $$$${values}\:{for}\:{a}_{{j}} .\:{When}\:{there}\:{are}\:\mathrm{101}\:{different}\: \\ $$$${numbers},\:{there}\:{exists}\:{atleast}\:{two},{say}\:{n}_{{i}_{{i}} } {and}\:{n}_{{i}_{\mathrm{2}} ,} \\ $$$${such}\:{that}\:{a}_{{j}_{\mathrm{1}} } ={a}_{{j}_{\mathrm{2}} } .\:{The}\:{larger}\:{of}\:{n}_{{i}_{\mathrm{1}} } \:{or}\:{n}_{{i}_{\mathrm{2}} } \:{must}\: \\ $$$${divide}\:{the}\:{other}\::\frac{{n}_{{i}_{\mathrm{1}} } }{{n}_{{i}_{\mathrm{2}} } }=\frac{\mathrm{2}^{{k}_{\mathrm{1}} } {a}_{{j}_{\mathrm{2}} } }{\mathrm{2}^{{k}_{\mathrm{2}} } {a}_{{j}_{\mathrm{1}} } }=\mathrm{2}^{{k}_{\mathrm{1}} −{k}_{\mathrm{2}} } ;\frac{{n}_{{i}_{\mathrm{2}} } }{{n}_{{i}_{\mathrm{1}} } }=\mathrm{2}^{{k}_{\mathrm{2}} −{k}_{\mathrm{1}} } \:. \\ $$
Commented by hardmath last updated on 29/Mar/24
Dear professor, Is this a generalization? How can we write in response?
$$ \\ $$Dear professor, Is this a generalization? How can we write in response?
Commented by A5T last updated on 29/Mar/24
It is not a generalization. Every n∈{1,2,3,...,200} is a product of the largest power of 2 that divides it and an odd number. For n≤200, such odd number ∈{1,3,5,7,...,199} with cardinality of 100. So,for 101 numbers, one must have two numbers with the same “odd part”. Say n_1 =2^k j,n_2 =2^q j If n_1 >n_2 , then n_2 ∣n_1 and if n_1 >n_2 , n_1 ∣n_2
$${It}\:{is}\:{not}\:{a}\:{generalization}.\:{Every}\:{n}\in\left\{\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{200}\right\} \\ $$$${is}\:{a}\:{product}\:{of}\:{the}\:{largest}\:{power}\:{of}\:\mathrm{2}\:{that}\:{divides} \\ $$$${it}\:{and}\:{an}\:{odd}\:{number}.\:{For}\:{n}\leqslant\mathrm{200},\:{such}\:{odd}\:{number} \\ $$$$\in\left\{\mathrm{1},\mathrm{3},\mathrm{5},\mathrm{7},…,\mathrm{199}\right\}\:{with}\:{cardinality}\:{of}\:\mathrm{100}. \\ $$$${So},{for}\:\mathrm{101}\:{numbers},\:{one}\:{must}\:{have}\:{two}\:{numbers} \\ $$$${with}\:{the}\:{same}\:“{odd}\:{part}''.\:{Say}\:{n}_{\mathrm{1}} =\mathrm{2}^{{k}} {j},{n}_{\mathrm{2}} =\mathrm{2}^{{q}} {j} \\ $$$${If}\:{n}_{\mathrm{1}} >{n}_{\mathrm{2}} ,\:{then}\:{n}_{\mathrm{2}} \mid{n}_{\mathrm{1}} \:{and}\:{if}\:{n}_{\mathrm{1}} >{n}_{\mathrm{2}} ,\:{n}_{\mathrm{1}} \mid{n}_{\mathrm{2}} \\ $$

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