Powers and Remainders [Number Properties] GMAT, GRE Quantitative Problem

z is a positive integer and multiple of 2; p = 4z, what is the remainder when p is divided by 10?

1. A)  10
2. B)  6
3. C)  4
4. D)  0
5. E)  It Cannot Be Determined
   
   
   
   
1. Every time, all the time, the remainder when an integer is divided by 10 is simply the units digit of that integer. To help see this, consider the following examples:
   14/10 is 1 with a remainder of 4
   5/10 is 0 with a remainder of 5
   105/10 is 10 with a remainder of 5
2. Also, z is a positive integer and is a multiple of 2. So, z must be a positive even integer.
3. We can rephrase the question to: "what is the units digit of 4 when raised to an even positive integer power?"
4. All integers raised to consecutive integer powers follow a repeating pattern. The units digit of 4 raised to an integer follows a specific repeating pattern:
   41 = 4
   42 = 16
   43 = 64
   44 = 256 
5. So we can say that4(odd number) --> units digit of 4 and 4(even number) --> units digit of 6
6. Since z must be an even integer, the units digit of p=4z will always be 6. Consequently, the remainder when p=4z is divided by 10 will always be 6.
   here are some examples:
   z=2 --> p=4z=16 --> p/10 = 1 with a remainder of 6
   z=4 --> p=4z=256 --> p/10 = 25 with a remainder of 6
   z=6 --> p=4z=4096 --> p/10 = 409 with a remainder of 6
   z=8 --> p=4z=65536 --> p/10 = 6553 with a remainder of 6

B