COOL divisors

Number Theory Level 4

How many divisors are divisible by ${ 10 }^{ 999 }$ but not with ${ 10 }^{ 998 }$

199 10 1000 1999

1 solution

Aman Gautam
Nov 29, 2014

${ 10 }^{ 999 }={ 2 }^{ 999 }{ 5 }^{ 999 }$
${ 10 }^{ 998 }={ 2 }^{ 998 }{ 5 }^{ 998 }$

total number of divisors = total no of divisors in ${ 2 }^{ 999 }{ 5 }^{ 999 }$ - total number of divisors in ${ 2 }^{ 998 }{ 5 }^{ 998 }$

therefore ${ 1000 }^{ 2 }-{ 999 }^{ 2 }$ = (1000 - 999)(1000 + 999) = 1999