A number theory problem by Baibhab Chakraborty

Just divide the whole problem into a couple of ideas and corollaries. 1) Whenever we talk about sum of digits, 9 often pings our mind. Any number divisible by 9 has its sum of digits divisible by 9. With short thinking, it can be found out that D(N)=remainder of the division of the number by 9. AS for example, D(52)=7 and the remainder of division of 52 by 9 is -2 or 7. 2) Here comes to our mind that what if the sum of digits is greater than 9? Well,the number so obtained is of form 9k+(remainder) - like in D(582)=15=9 X 1+6 and the remainder of division by 582 is also 6! 3) If we apply a series of this function to a great number the end number is the remainder itself. Check it for D(5899). Now lets proceed to the question. First of all- Remainder of 4444^{4444} divided by 9=??? One important logic applied is that R(N^{y}) divided by x =R(R(N divided by x))^{y} divided by x. Like the remainder of 28^{4} divided by 9 is remainder of (remainder of 28 divided by 9)^{4}, divided by 9. :/ What divide and divide! remainder of 28^{4} divide by 9 is 1 and , remainder of 28 divided by 9 is 1 and 1^4=1 and further division by 9 leaves remainder 1.

For our problem, Remainder of 4444^{4444} divided by 9 =Rem.( $\frac{(remainder of 4444 divided by 9)^{4444}}{9}$ ) =Rem.( $\frac{7^{4444}}{9}$ ) Now by cyclic patterns, 7^{1}/9--7 7^{2}/9--4, 7^{3}/9--1 and so cycle repeats itself after 3 . So remainder of 7^{4444} is 7. * * * * * * * * * Remainder of 4444^{4444} divided by 9 is 7 * * * * * * * *

Now as we learnt at any stage of any number of D(N) functions applied, the answer is of form 9k+7. Now our thought is how , how we understand what is value of only D(4444^{4444})? So lets try logarithms. Number of digits in N=4444^{4444} is =log(N)+1. =4444 log(4444)+1. AT LEAST IT WOULDNT EXCEED THE NUMBER OF DIGITS IN 4444 log(10^{4}), i.e, 17776. THUS D(N) WOULDN'T EXCEED 17776 X 9=159984( as it is the maximum possible sum of digits of a 17776 digit number; all 9s) THUS D(N)<159984 D(D(N))<D(159984)<D(999999)=54 D(D(N))<54 D(D(D(N)))<D(54)=9.......and we know the answer is to be in form 9k+7, i.e, 7 or 16 or 25 or........ * * * * * * * * * ANSWER IS 7 * * * * * * * * * * * * * . Hope this helps :)