Question Number 48897 by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18
$${find}\:{the}\:{highest}\:{power}\:{of}\:\mathrm{5}\:{contained}\:{in}\mathrm{158}!\:\: \\ $$
Answered by mr W last updated on 29/Nov/18
$$\mathrm{155}=\mathrm{31}×\mathrm{5} \\ $$$$\mathrm{5},\mathrm{2}×\mathrm{5},\mathrm{3}×\mathrm{5},...,\mathrm{31}×\mathrm{5}\Rightarrow\mathrm{31}\:{times} \\ $$$$\mathrm{1}×\mathrm{5},\mathrm{2}×\mathrm{5},\mathrm{3}×\mathrm{5},...,\mathrm{6}×\mathrm{5}\Rightarrow\mathrm{6}\:{times} \\ $$$$\mathrm{1}×\mathrm{5}\Rightarrow\mathrm{1}\:{time} \\ $$$${totally}\:\Rightarrow\mathrm{38}\:{times}\:\mathrm{5}\:{in}\:\mathrm{158}! \\ $$$${i}.{e}.\:\mathrm{5}^{\mathrm{38}} \:{is}\:{contained}\:{in}\:\mathrm{158}! \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 30/Nov/18
$${thank}\:{you}\:{sir}... \\ $$
Answered by rahul 19 last updated on 29/Nov/18
![[((158)/5)]+[((158)/(25))]+[((158)/(125))]= 31+6+1=38 ⇒5^(38) is contained in 158!](Q48908.png)
$$\left[\frac{\mathrm{158}}{\mathrm{5}}\right]+\left[\frac{\mathrm{158}}{\mathrm{25}}\right]+\left[\frac{\mathrm{158}}{\mathrm{125}}\right]=\:\mathrm{31}+\mathrm{6}+\mathrm{1}=\mathrm{38} \\ $$$$\Rightarrow\mathrm{5}^{\mathrm{38}} \:{is}\:{contained}\:{in}\:\mathrm{158}! \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 30/Nov/18
$${good}...{thank}\:{you}\:{rahul}... \\ $$