s is a positive integer and multiple of 2; p = 4^s, what is the remainder when p is divided by 10? Answers
A) 10  B) 6  C) 4  D) 0  E) it can not be determined

Rationale: 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 s is a positive integer and multiple of 2 . any integer that is a multiple of 2 is an even number . so , s 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 s must be an even integer , the units digit of p = 4 ^ s will always be 6 . consequently , the remainder when p = 4 ^ s is divided by 10 will always be 6 . in case this is too theoretical , consider the following examples : s = 2 - - > p = 4 ^ z = 16 - - > s / 10 = 1 with a remainder of 6 s = 4 - - > p = 4 ^ z = 256 - - > s / 10 = 25 with a remainder of 6 s = 6 - - > p = 4 ^ z = 4096 - - > s / 10 = 409 with a remainder of 6 s = 8 - - > p = 4 ^ z = 65536 - - > s / 10 = 6553 with a remainder of 6 answer : b .
Answer: B