<p>What is the remainder when 2 ^400 is divided by 10 ?
ans-
0
2
4
6
8</p>
<p>ans=6
This answer was found out by using modular arithmetic. Is there a faster method to do it?</p>
<p>I think modular arithmetic would be the best approach for this problem. It would only take 1 minute at most. Probably around 30-40 seconds.</p>
<p>Here’s a nice quick way:</p>
<p>2^400 = (2^4)^100 = 16^100. Now simply observe that whenever you multiply 2 integers together that each end in a 6, the resulting integer also ends in a 6. Thus 16^100 ends in a 6.</p>
<p>wow…! DrSteve… You’re amazing!</p>
<p>Really nice logic DrSteve amazing.</p>
<p>A method that is perhaps best suited to the SAT is to observe that a pattern emerges when dividing 2^k by 10 and looking at the remainder.</p>
<p>2^1 → 2, 2^2 → 4, 2^3 → 8, 2^4 → 6</p>
<p>2^5 → 2, 2^6 → 4, etc.</p>
<p>The pattern is 2,4,8,6, 2,4,8,6, etc. The 400 hundred’th term is 6 – i.e. same as the 4th term.</p>