What is tens digit at 2^642? S

Question

What is tens digit at 2^642?
S

Dookie · Answer

Are you sure the question asks for the tens digit?
If it was the units digit i think the answer should be "4".
What's the answer you have ?

saurya_s · Answer

Dookie, haven't got the answer. yes it is tens digit

hardworker_indian · Answer

Is the answer 8?
A weird method to explain. If OA matches, I will surely explain.

twixt · Answer

I spent 5 minutes on this one and and did not find any trick.
Btw Dookie could you please explain how you obtained 4 as a units digit ?
Maybe there is something wrong with 642 which contains the prime 107...

Dookie · Answer

I was looking for the units digit. 
Noticed that the units digit changes in a sequence of 4: 
2,4,8,6,2,4,8,6,2,4,8,6....
642/4 = 160 + Remainder 2.
So the units digit should be "4".

twixt · Answer

I ve got the trick for the units digit... just a serie 2 4 8 6 2 4 8 6... I think there is the same, more complex, approach to conduct for the tens digit... I com back with the answer ;-)

twixt · Answer

Finally I obtain 4 as a units digit and 8 for the tens digit. For units, 2 <=> 2^5 642=2+5*128 For tens, 2 <=> 2^20 642=2+20*32

myfrankenstein · Answer

I have a longish approach to solving this problem that involves successive breakdowns.
642 = 2*3*107
2 ^ 642 = (60+4) ^ 107 = (60 + 4) ^ 107.
(Now the critical part) Considering a binomial expansion in all binomial terms the only term that will affect the ten's digit or add to it shall be 4^107. The rest have 0 at ten's digit.
4^107 = (4^2)*(4^105) = (16) *    
Need to consider only 16 * (4 ^21) = 16* 4 * (4^20)
= (60+4) * (1020 + 4 )^4
Again consider only 4 * (4^4 ) 
4* (250 + 6) => last 2 digits r like 24 

We can say for sure that 4 is the unit's digit and 2 the ten's digit. We can't be say anything about the hundred's digit as we eliminated only considering the ten's digit.
Instead of 60 + 4 yu can use 1020+ 4 or any other combination as well. This involves using binomial expansion to eliminate non considerants.
Unit's digit : 4
Ten's digit : 2
Time taken 130 seconds.

venksune · Answer

Just a quick tip to calculate the unit digits of a crazy exponential number as in the case here.
Dookie has explained well the pattern method. There is a shorter trickier method.

Case1: When  2,3, 7 or 8  are the digits in the unit place of the base number, then divide the power of that number by 4 and replace the power of the remainder 1,2,3, or 0. In the problem under discussion, we will divide 642 by 4 (as 2 is the unit digit of the base number). We have remainder as 2. So, the unit digit of 2^642 is 2^2 = 4.

Case 2: When  0,1, 5 or 6  are the digits in the unit place of the base number, then for all values of the power, the number has the same digit in the unit place. Say. 5^322 will be 5.

Case 3: When  4  is in the unit place of the base number, then the unit digit will contain 4 or 6 according to odd or even power respectively.

Case 4: When  9  is in the unit palce of the base number then the unit digit will contain 9 or 1 in the unit place according to the odd power or even power respectivey. (ex. 9^1= 9 , 9^2 =8 1 , 9^3=72 9 ...)

myfrankenstein · Answer

The repetition is based on recognizing the pattern and works only for the unit's digit.
You r better off understanding the pattern and analysis rather than the rule.

venksune · Answer

Franky,
The rule is a derivative of the pattern approach and analysis. Yes, its only for the units digits.

saurya_s · Answer

Franky, your approach is right. What is the basis of selecting 4^107 term. Terms affecting tens digit should have only one zero (terms which won't afect tens digit are those with at least two zeros at end) I guess on that basis you should also consider the term where 60 occurs one. Can you please explain it what I am missing.
S

amernassar · Answer

2^642=18249762470488780874564686422801165299572914028994239722316770071597100668834709546023651245269485599114569238294377629242754818885501751993010645278888856753007978697441059800331496768986415104

that elimanates the idea that the tenth digit is 2.


Note: I didn't calculate it myself. (found the answer in 5 sec using Mathematica)

myfrankenstein · Answer

You r right about the hundred's digit consideration. This would make it too tedious then. I made a mistake on ten's digit . We can get the answer using the approach above...only that it will take too long

Either there is another shortcut or guessing is the best way forward.

What is tens digit at 2^642? S

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