For how many natural numbers x, is the expression: (x ² + 2x + 3) divisible by 35 ?

(In reply to re(2): Solution by Charlie)

Well, if you added 1 to each, you would end up with values of 1,2,3,4,5; 5 is the "0" value.

What Charlie is saying is this: Replace x with 5y+z, where y and z are non-negative integers, with z less than 5. This gives 25y² +10yz + z² + 10y+ 2z + 3 or 5(5y²+2yz+2y) + z² + 2² + 3 . Then divide that by 5 to get what it is mod 5, which is easily seen to be what we started with, only with z to be 0 through 4. So using higher forms of the same number base 5 (3,8,13,18...) will be the same for each one.

Posted by Gamer on 2004-01-06 20:24:43