Question Number 200200 by Fridunatjan08 last updated on 16/Nov/23
$${Find}\:{four}\:{positive}\:{integers}, \\ $$$$\:{each}\:{not}\:{exceeding}\:\mathrm{70000}\:{and}\: \\ $$$${each}\:{having}\:{more}\:{than}\:\mathrm{100} \\ $$$$\:{divisors}. \\ $$
Answered by mr W last updated on 16/Nov/23
$${examples}: \\ $$$$\mathrm{50400}\:{has}\:\mathrm{108}\:{divisors} \\ $$$$\mathrm{55440}\:{has}\:\mathrm{120}\:{divisors} \\ $$$$\mathrm{60480}\:{has}\:\mathrm{112}\:{divisors} \\ $$$$\mathrm{65520}\:{has}\:\mathrm{120}\:{divisors} \\ $$$$\mathrm{69300}\:{has}\:\mathrm{108}\:{divisors} \\ $$$$...... \\ $$
Commented by mr W last updated on 17/Nov/23
$${see}\:{also}\:{explanation}\:{from}\:\mathrm{MM42}\:{sir} \\ $$$${in}\:{Q}#\mathrm{200236} \\ $$
Commented by Fridunatjan08 last updated on 16/Nov/23
$${but}\:{how}\:{to}\:{find}\:{these}\:{numbers}? \\ $$
Commented by mr W last updated on 17/Nov/23
$${the}\:{number}\:{can}\:{be}\:{expressed}\:{as} \\ $$$$\mathrm{2}^{{a}} ×\mathrm{3}^{{b}} ×\mathrm{5}^{{c}} ×\mathrm{7}^{{d}} ×\mathrm{11}^{{e}} ×... \\ $$$${now}\:{you}\:{select}\:{a},{b},{c},{d},{e},...\:{such}\:{that} \\ $$$$\mathrm{2}^{{a}} ×\mathrm{3}^{{b}} ×\mathrm{5}^{{c}} ×\mathrm{7}^{{d}} ×\mathrm{11}^{{e}} ×...<\mathrm{70000}\:{and} \\ $$$$\left({a}+\mathrm{1}\right)\left({b}+\mathrm{1}\right)\left({c}+\mathrm{1}\right)\left({d}+\mathrm{1}\right)\left({e}+\mathrm{1}\right)...>\mathrm{100} \\ $$$${we}\:{must}\:``{error}\:{and}\:{try}''! \\ $$