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

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