q is a positive integer and multiple of 2; p = 4^q, what is the remainder when p is divided by 10? Answers
A) 10
B) 6
C) 4
D) 0
E) It Cannot Be Determined
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 q is a positive integer and multiple of 2. Any integer that is a multiple of 2 is an even number. So, q 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) --&gt; units digit of 4   4^(even number) --&gt; 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 q must be an even integer, the units digit of p=4^q will always be 6. Consequently, the remainder when p=4^q is divided by 10 will always be 6.   In case this is too theoretical, consider the following examples:   q=2 --&gt; p=4^q=16 --&gt; p/10 = 1 with a remainder of 6   q=4 --&gt; p=4^q=256 --&gt; p/10 = 25 with a remainder of 6   q=6 --&gt; p=4^q=4096 --&gt; p/10 = 409 with a remainder of 6   q=8 --&gt; p=4^q=65536 --&gt; p/10 = 6553 with a remainder of 6   Answer: 6.
<result>B</result>
q is a positive integer and multiple of 2; p = 4^q, what is the remainder when p is divided by 10?