Problem Solving

What's the remainder when (4444)^(4444) is divided by 9?


Remainder when 7^0+7^1+7^2......+7^25 is divided by 14?

MBA.Aspirant wrote:What's the remainder when (4444)^(4444) is divided by 9?


Remainder when 7^0+7^1+7^2......+7^25 is divided by 14?

1]This is for the first question.
For such questions, I usually want to see if there is any pattern. Remainder of[(4444^1)/9] =7
Remainder of[ (4444^2)/9]=4
Remainder of [(4444^3)/9]= 1
Remainder of [(4444^4)/9]= 7
see that? after the 3rd, it follows a similar pattern. Though finding the remainders appears too cumbersome, it shall take you a minute or less to do so if you really understand the properties of remainders. The following formula is useful in this regard:
Remainder[(m * n)/d] = Remainder[m/d] * Remainder[n/d] [I call this the multiplication rule of remainders, this is only for my purpose] N.B.Here if you get a value greater than d keep on dividing it by d till you get a value less than d, and that value is the remainder.

Getting back to the solution. See the pattern above closely. Excluding the first, the remainder is 7 for every 3rd. value. So getting back to our original question, we need to find for the 4444th. pattern. 4443 is divisible by 3; so the remainder for the 4443th pattern is 7. Therefore, looking at the pattern, the reminder for the 4444th. pattern shall be 4.

So the answer is 4.

2] This is my solution to the second question.
When working with remainder problems, the following rule may, at times, be handy.
Remainder[(X + Y)/z] = Remainder(X/z)+ Remainder (Y/z)[ I call this the addition rule of remainders, that is, once again, because I don't know the name of the rule ]
In like vein, your question can be expressed as:
Remainder[(7^0 +7^1 +...)/14] = Remainder(1/14)+ Rema.[(7^1)/14]...
= 1 +7 +7 +7...[ you have 25 such sevens]
Here it is helpful to note that for any integer x>1 and where a is dividend, d is a divisor and a>d, (a^x)/d leaves the same remainder.
Getting back to the solution, 1+7+7+7...[25 such sevens] =176[ Had this been a number less than 14, we would have taken it as an answer;however as in this case it is greater than 176, we have to divide 176 by 14 and find the remainder. That way we get 8 as a remainder.
Hope this helps.