if n is a positive integer, what is the remainder when ( 4 ^ ( 4 n + 3 ) ) ( 6 ^ n ) is divided by 10?  Choices:
A) 1	B) 2	C) 4	D) 6	E) 8
<gadget id="calculator">3 * 3</gadget>
<output>9</output>

<gadget id="calculator">9 * 6</gadget>
<output>54</output>

<gadget id="calculator">54 % 10</gadget>
<output>4</output>

<result>C</result>
if n is a positive integer, what is the remainder when ( 4 ^ ( 4 n + 3 ) ) ( 6 ^ n ) is divided by 10?
reminder(multiply(multiply(const_3, const_3), 6), 10)
"this one took me bout 3 1 / 2 min . just testin numbers and what not . first notice that n is positive . save time by noticing thati worked out one solution where n = 0 only to find that thats not an option : p . 1 - 7 stands for ^ 1 thru 7 1 : 7 * 1 = 7 2 : 7 * 7 = 9 3 : 7 * 9 = 3 4 : 7 * 3 = 1 5 : 7 * 1 = 7 6 : 7 * 7 = 9 7 : 7 * 9 = 3 pattern repeats every @ 5 . notice every ^ 4 or multiple of 4 is always going to be 1 . this is just for future notice for similar problems . so 7 ^ 4 n + 3 - - - > if n = 1 then its ( ( 7 ^ 7 ) * 6 ) ) / 10 which can say is going to be 3 * 8 - - > 18 / 10 - - > r = 8 now from here if id double check just to make sure . 7 ^ 4 ( 2 ) + 3 * 6 ^ 2 - - - > 7 ^ 11 * 36 or we can just say again 7 ^ 11 * 6 ( b / c we are only interested in the units digit ) . since ^ 12 is going to be 1 that means ^ 11 is going to be 3 ( as taken from our pattern ) so again 3 * 6 = 18 / 10 - - - > r = 8 . c or j in this problem ."