# Practice GMAT Problem Solving Question

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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:

4^{1}= 4

4^{2}= 16

4^{3}= 64

4^{4}= 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

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