Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

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

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

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

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: 75 has 6 factors: 1,3,5,15,25,75.

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

Commented by math1967 last updated on 01/Jul/18

(x+1)(y+1)−1=75 ∴(x+1)(y+1)=76

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

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

not correct sir.

$${not}\:{correct}\:{sir}. \\ $$

Commented by math1967 last updated on 01/Jul/18

2^4 ×3^3 ×75=32400?

$$\mathrm{2}^{\mathrm{4}} ×\mathrm{3}^{\mathrm{3}} ×\mathrm{75}=\mathrm{32400}? \\ $$

Commented by MrW3 last updated on 01/Jul/18

that′s correct sir.

$${that}'{s}\:{correct}\:{sir}. \\ $$

Commented by MrW3 last updated on 01/Jul/18

For more see original question in Q11075.

$${For}\:{more}\:{see}\:{original}\:{question}\:{in} \\ $$$${Q}\mathrm{11075}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com