gijoedude wrote:
Answer is 7.
Step 1: 3^3= 27.
Step 2: Therefore 3^(3^3)=3^27. = 3*3*3*.......3 (27 3s).
Step 3: We know 3*3*3 = 27
Step 4: Hence step 2 can be simplified as 27*27*...27 (9 27s)
Step 5: We know 27*27*27 ends in 3 (multiply it out. No need to do the entire multiplication. Only the last digits matter).
Step 6: Hence we end up abc3*abc3*abc3 = a number ending with 7 (again only the last digits matter)
Step 7: Voila!
This is way too complicated !
Every digit has a pattern that repeats. See the chart given below. Some repeat in multiples of 4, others repeat in multiples of 1 and 2.
Key takeaway is that EVERY POWER REPEATS !
Powers of 2 -
2,4,8,6,,2,4,8,6
Powers of 3-
3,9,7,1,3,9,7,1
Powers of 4 -
4,6,4,6,4,6Powers of 5 -
5,5,5,5,5,5,5,
Powers of 6 -
6,6,6,6,6,6,6,
Powers of 7 -
7,9,3,1,7,..
Powers of 8 -
8,4,2,6,8
Powers of 9 -
9,1,9,1
They key is to find the pattern for reptition. A units digit for a power of 3 repeats every 4 terms.
The units digit for 3 ^ 27 ==> Hence the term that corresponds to Remainder of (27 divided by 4) = 3rd term = 7 is the answer
To stretch this a lil bit more
2 ^ 100 = Remainder of (100 divided by 4) = Remainder = 0 , ie 4th term
A word of caution - Focus on the word remainder.
The division only tells us the
POSITION of the term in the sequence and NOT the repeating term itself.
2^50 =->
REMAINDER OF (50 divided by 4) =
2ND TERM in the sequence, hence 4
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