Challenge Mathematics problem? #11?

[I need help]

Suppose you sum the digits of integer x, then sum the digits of your answer, then sum the digits of that answer, and so on until you reach a single digit number. This single digit number is called the digital root of x. [For example, the digital root of 25,009,010 is 8.]
A)Find the largest number less than a billion that has digital root 4.
B)Find the digital root of 2^(123,456,789,010)

[thomasoa]

The "digital root" of a number is always the remainder of the number when divided by 9 (except when the number is divisible by 9, then the digital root will be 9.)

Basically, if x is your number, and s(x) is the sum of its digits, then x-s(x) is always divisible by 9.

In the first problem, then, you want the largest number less than 1,000,000,000 which has 8 as a remainder when divided by 9, which is: 999,999,998.

The second problem will use that 2^6=1 (mod 9)

123456789006 is divisible by 6, so:

2^123456789010 = 2^4 (mod 9)
2^4=16 = 7 (mod 9)

So its digital root is 7.