Thread: elementary number theory: sum of digits

let x be an integer and let j be the sum of the digits of x. Subtract j from x and get x-j=y. Let k be the sum of the digits of y. Show that y-77,778j cannot equal 7,777,777.

I have tried to do this with divisibility criteria. For instance 77,778j must be even and 77,778 is divisible by 3 while 7,777,777 is not but the arbitrariness of x and y leave me stuck.

Actually, y must be divisible by three by the following theorem: Let x be an integer and j be the sum of its digits in base 10. Then no matter what. Therefore 3|y=x-j, so 3|y-77778j and 7777777 does not.

Pf: Let x be an integer. Expressed in base 10, for some unique set of integers . Since , for all k, so . QED

Media_Man, you said that y must be divisible by 3, in other words, 3 divides y, which is correct ... but later on, you write that y divides 3