Determine all three-digit numbers N having the property that N is divisible by 11, and N/11 is equal to the sum of the squares of the digits of 11.
Anyone know how to solve this one. We write N as abc then...
Since N is divisible by 11 we know that a - b + c is divisible by 11. So a-b+c = 0 or 11.
Then we also know that
I can't get any sort of 'nice' solution from this and just end up with having to check 18 cases. (a+c=b, sub into above than a=0,1,2,... etc)
Maybe there is another method which is better ...
the actual value of is
which leads to two cases and
Let's check the first one :
Since substitute it we obtain
which is a quadratic equation of , check its discriminant and it should be a square ie:
( Note we neglect the case because later we will find that must not be an integer . )
Therefore , the two possible ways to write in the form are :
and we have or the only solution is and ( corresponding to )
so or which gives
Therefore , the required number is
Use the similar method to see if there is another solution in the second case . It should be .