Question Number 200236 by mr W last updated on 16/Nov/23
$${what}\:{is}\:{the}\:{smallest}\:{natural}\:{number} \\ $$$${which}\:{has}\:{at}\:{least}\:\mathrm{100}\:{divisors}? \\ $$
Answered by MM42 last updated on 16/Nov/23
$$\mathrm{2}^{\mathrm{4}} ×\mathrm{3}^{\mathrm{4}} ×\mathrm{5}×\mathrm{7}=\mathrm{45360}\:\checkmark \\ $$
Commented by Rasheed.Sindhi last updated on 16/Nov/23
$$\mathcal{H}{ow}\:{to}\:{prove}\:{it}? \\ $$
Commented by MM42 last updated on 17/Nov/23
$$\mathrm{100}=\mathrm{2}×\mathrm{2}×\mathrm{5}×\mathrm{5} \\ $$$${n}=\mathrm{2}^{{a}} ×\mathrm{3}^{{b}} ×\mathrm{5}^{{c}} ×\mathrm{7}^{{d}} \\ $$$$\Rightarrow{D}_{{n}} =\left({a}+\mathrm{1}\right)\left({b}+\mathrm{1}\right)\left({c}+\mathrm{1}\right)\left({d}+\mathrm{1}\right)=\mathrm{100} \\ $$$${we}\:{give}\:{the}\:{largest}\:{power}\:{to}\:{the}\: \\ $$$${smallest}\:{base}\: \\ $$
Commented by Rasheed.Sindhi last updated on 16/Nov/23
$$\mathbb{T}\boldsymbol{\mathrm{han}}\Bbbk\boldsymbol{\mathrm{s}}\:\boldsymbol{\mathrm{sir}}! \\ $$
Commented by mr W last updated on 17/Nov/23
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