What is the smallest integer greater than 0 that can be written entirely with zeros and ones and is evenly divisible by 225?
The prime factorization of 225 is 3*3*5*5
Accordingly, 225 can be written as the product of two positive integers a and b with gcd(a,b) =1 in the following manner:
225=9*25 ..........(i)
Now, a number is divisible by 9 if and only if the sum of its digits is divisible by 9.
Therefore, the smallest number consisting of all 1's that is divisible by 9 must be
111,111,111 .........(ii)
Again, the last two digits of multiples of 25 cycles as 25,50,75,00 which repeats indefinitely....... (iii)
From (i), (ii) and (iii), we must have the required minimum positive integer as:
111,111,111*100 =11,111,111,100
Edited on April 5, 2022, 10:39 am
Solution
