June 29, 2005

Raised To Raised To Raised To

June 29, 2005 Posted by Vijay , 3 comments
That precisely sums up this problem which was also posted on ascent4cat.

What is unit digit for 7 exp 11 exp 22 exp 33 ?

i.e., 7 raised to 11 raised to 22 raised to 33 .. phew ..!!

And, the answer so simply put by one indra goel as follows :)

The powers of 7 have the following unit places (starting from 7^1) 7,9,3,1,7,9... so at every 5th power there is repetition, therefore we can remove all the 5's from the power terms(power mod 5) and see the remainder to get the units place...In this problem 7 exp 11 exp 22 exp 33..11 mod 5=1..22 mod 5=2..33 mod 5=3..multiplying the power remainders(1*2*3) we get 6 which further gives 6 mod 5=1 thus finally we have the number 7 in the units place as per the pattern we generated

Good one .. :)

I LOVE numbers :)


Aslan said...

me too! :)

Vijay said...


We know that any power of 11 will end in 1. Hence we need not consider the other powers of 11 here at all. Knowing that its every 5th power for which there is a repetition, we can safely say, the end result of 7 exp 11 exp 22 exp 33 has to be something like a 7 exp(x1), where x can be any number. Hence, it is of the form 7 exp 1. Since 5 mod 1 is 1, 7 exp x1 has to end in 7 .Hence the answer, 7.

Vijay said...

Hey guys, why don't you correct me when I am wrong :(
The answer should be 9. The product of 11^22^33 will definitely end in 1, but the division, owing to the cycle of 4 for the remainder, has to be by 4 and NOT by FIVE as previously mentioned.In that case, my alternate theory will go for a toss, coz, 31/4 has remainder = 3, while 41/4 has a remainder 1, which leaves my alternate quite ambiguous. Indra's method is correct but the division should be by 4 and NOT 5. In that case, we have , division of the product of 11,22 and 33 which gives 2 and the answer , 9.