Question and Answers Forum

All Questions      Topic List

Number Theory Questions

Previous in All Question      Next in All Question      

Previous in Number Theory      Next in Number Theory      

Question Number 11606 by Joel576 last updated on 29/Mar/17

What is the smallest positive integer x for  which (1/(32)) = (x/(10^y ))  for some positive integer y ?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{smallest}\:\mathrm{positive}\:\mathrm{integer}\:{x}\:\mathrm{for} \\ $$$$\mathrm{which}\:\frac{\mathrm{1}}{\mathrm{32}}\:=\:\frac{{x}}{\mathrm{10}^{{y}} }\:\:\mathrm{for}\:\mathrm{some}\:\mathrm{positive}\:\mathrm{integer}\:{y}\:? \\ $$

Answered by mrW1 last updated on 29/Mar/17

32x=10^y =(2×5)^y =2^y ×5^y   2^5 x=2^y ×5^y   x=2^(y−5) ×5^y   for x to be positive integer, it must be  y−5≥0 or y≥5  ⇒min. x=2^0 ×5^5 =3125

$$\mathrm{32}{x}=\mathrm{10}^{{y}} =\left(\mathrm{2}×\mathrm{5}\right)^{{y}} =\mathrm{2}^{{y}} ×\mathrm{5}^{{y}} \\ $$$$\mathrm{2}^{\mathrm{5}} {x}=\mathrm{2}^{{y}} ×\mathrm{5}^{{y}} \\ $$$${x}=\mathrm{2}^{{y}−\mathrm{5}} ×\mathrm{5}^{{y}} \\ $$$${for}\:{x}\:{to}\:{be}\:{positive}\:{integer},\:{it}\:{must}\:{be} \\ $$$${y}−\mathrm{5}\geqslant\mathrm{0}\:{or}\:{y}\geqslant\mathrm{5} \\ $$$$\Rightarrow{min}.\:{x}=\mathrm{2}^{\mathrm{0}} ×\mathrm{5}^{\mathrm{5}} =\mathrm{3125} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com