Refreshing-an-old-question-Find-the-smallest-number-which-is-a-multiple-of-75-and-has-75-divisors-

Question Number 38933 by MrW3 last updated on 01/Jul/18

$$\left[{Refreshing}\:{an}\:{old}\:{question}\right] \\ $$$${Find}\:{the}\:{smallest}\:{number}\:{which}\:{is} \\ $$$${a}\:{multiple}\:{of}\:\mathrm{75}\:{and}\:{has}\:\mathrm{75}\:{divisors}. \\ $$

Commented by math1967 last updated on 01/Jul/18

75=3×5^2 ∴factor of 75 (1+1)(2+1)−1=5 let3^x ×5^y has 75 factor (x+1)(y+1)=76=19×4 for smallest multiple of 75 x=18 y=3 ∴required number3^(18) ×5^3 Is it correct?

$$\mathrm{75}=\mathrm{3}×\mathrm{5}^{\mathrm{2}} \:\therefore{factor}\:{of}\:\mathrm{75} \\ $$$$\left(\mathrm{1}+\mathrm{1}\right)\left(\mathrm{2}+\mathrm{1}\right)−\mathrm{1}=\mathrm{5} \\ $$$${let}\mathrm{3}^{{x}} ×\mathrm{5}^{{y}} \:{has}\:\mathrm{75}\:{factor} \\ $$$$\left({x}+\mathrm{1}\right)\left({y}+\mathrm{1}\right)=\mathrm{76}=\mathrm{19}×\mathrm{4} \\ $$$${for}\:{smallest}\:{multiple}\:{of}\:\mathrm{75} \\ $$$${x}=\mathrm{18}\:\:\:\:\:{y}=\mathrm{3}\:\therefore{required}\:{number}\mathrm{3}^{\mathrm{18}} ×\mathrm{5}^{\mathrm{3}} \\ $$$${Is}\:{it}\:{correct}? \\ $$

Commented by MrW3 last updated on 01/Jul/18

Not yet, but you are approaching the goal. Why 76? The requested number should have exactly 75 factors. Besides an other hint: 2×75 is smaller than 3×75, and both are a multiple of 75!

$${Not}\:{yet},\:{but}\:{you}\:{are}\:{approaching}\:{the} \\ $$$${goal}.\:{Why}\:\mathrm{76}?\:{The}\:{requested}\:{number} \\ $$$${should}\:{have}\:{exactly}\:\mathrm{75}\:{factors}.\:{Besides} \\ $$$${an}\:{other}\:{hint}:\:\mathrm{2}×\mathrm{75}\:{is}\:{smaller}\:{than} \\ $$$$\mathrm{3}×\mathrm{75},\:{and}\:{both}\:{are}\:{a}\:{multiple}\:{of}\:\mathrm{75}! \\ $$

Commented by MrW3 last updated on 01/Jul/18

$${Hint}:\:\mathrm{75}\:{has}\:\mathrm{6}\:{factors}:\:\mathrm{1},\mathrm{3},\mathrm{5},\mathrm{15},\mathrm{25},\mathrm{75}. \\ $$

Commented by math1967 last updated on 01/Jul/18

$$\left({x}+\mathrm{1}\right)\left({y}+\mathrm{1}\right)−\mathrm{1}=\mathrm{75} \\ $$$$\therefore\left({x}+\mathrm{1}\right)\left({y}+\mathrm{1}\right)=\mathrm{76} \\ $$

Commented by math1967 last updated on 01/Jul/18

(x+1)(y+1)=75 x=24 y=2 3^(24) ×5^2 pls check

$$\left({x}+\mathrm{1}\right)\left({y}+\mathrm{1}\right)=\mathrm{75} \\ $$$${x}=\mathrm{24}\:{y}=\mathrm{2} \\ $$$$\mathrm{3}^{\mathrm{24}} ×\mathrm{5}^{\mathrm{2}} \:{pls}\:{check} \\ $$

Commented by MrW3 last updated on 01/Jul/18