Practice GMAT Problem Solving Question

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

Correct Answer: B

It is essential to recognize that 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:
4/10 is 0 with a remainder of 4
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
It is also essential to remember that the z is a positive integer and multiple of 2. Any integer that is a multiple of 2 is an even number. So, z must be a positive even integer.
With these two observations, the question can be simplified to: "what is the units digit of 4 raised to an even positive integer?"
The units digit of 4 raised to an integer follows a specific repeating pattern:
4¹ = 4
4² = 16
4³ = 64
4⁴ = 256
4^{(odd number)} --> units digit of 4
4^{(even number)} --> units digit of 6
There is a clear pattern regarding the units digit. 4 raised to any odd integer has a units digit of 4 while 4 raised to any even integer has a units digit of 6.
Since z must be an even integer, the units digit of p=4^z will always be 6. Consequently, the remainder when p=4^z is divided by 10 will always be 6.
In case this is too theoretical, consider the following examples:
z=2 --> p=4^z=16 --> p/10 = 1 with a remainder of 6
z=4 --> p=4^z=256 --> p/10 = 25 with a remainder of 6
z=6 --> p=4^z=4096 --> p/10 = 409 with a remainder of 6
z=8 --> p=4^z=65536 --> p/10 = 6553 with a remainder of 6