Question 1066754: find the smallest number which is grater than 111,111,000 and divisible by 8 and 9
Answer by KMST(5328)   (Show Source):
You can put this solution on YOUR website! THE IDEA:
To be divisible by the relatively prime numbers  and  ,
a number has to be divisible by 
THE HARD WAY:
Dividing  by 72,
we get  as the quotient,
with  as a remainder,
so  is less than ,
but the next multiple of  is
 .
EASIER:
Dividing by 72,
we get as the quotient,
with as a remainder,
so adding  to
we get the next multiple of :
 .
WITHOUT DIVIDING:
Like all numbers ending in  ,
is divisible by ,
because  is divisible by .
When dividing by ,
the remainder can be found by adding the digits,
and repeating the digit adding with the result,
as many times as needed until you get a single digit result.
The final result is the remainder, unless it is .
If the final result is ,
then the number is divisible by ,
and the remainder is  .
The sum of the digits of is  ,
so when dividing by ,
we get a quotient  , and the remainder is .
So,  is a multiple of ,
and it is even a multiple of  ,
but it is not a multiple of .
If I add a multiple of that is also a multiple of ,
such as  or  ,
the sum will also be a multiple of and a multiple of .
One of those sums must be a multiple of too.
 is not a multiple of ,
because adding digits we get
 and  ,
but is a multiple of ,
because  and  .
|
|
|
