The units digit of (137^13)^47 is:

Question

(A) 1
(B) 3
(C) 5
(D) 7
(E) 9

Kudos for a correct  solution .

Turkish · Accepted Answer

The units digit of (137^13)^47 is:

The  unit  digit when 7 to power of 1, 2,3,4 is

7^1=7
7^2=9 
7^3=3
 7^4=1 
7^5=7
7^6=9
that means the cycle repeats after 4th power

(137^13)^47 = (we have to multiply the powers so 13*47=611 Now we have 137^611. If we divide the power 611 by 4 (coz of the cycle repeats after 4) we have 611/4 =152 and has a reminder  3 . Which means at 152 the unit digit will be at 7^4=1 or 1. the reminder of 3 will bring it to 7^3=3 or 3. So, the unit digit will be 3

So in 137^611 the unit digit will be 3

mehrdadtaheri92 · Answer

ANSWER is B NOT A,

Here is why the answer is B....

Number 7 has 4 cycle in its repeating power :

7^1 =7

7^2= ..9

7^3 = ...3

7^4= ....1

from this point the unit digit of 7^5 is 7 so it is repeating so keep in mind that we have 4 cycles that are repeating ...

so here lets apply this to the problem, first look at the initial term without considering the power 47 : ( 137^13 ) , here ONLY unit digit matters so we have : ( 7^13 ) by applying above cycle

pattern , we see that 13 is 3*4 +1 , so we should cover 4 FULL cycle and the 13th term is our answer . by viewing to the cycle we see that after 4 full cycle the 13th number is conform to the first number in 7 power so 13th's unit digit is 7...

Now lets apply the power 47 ... we have a number some thing like this : (.....7 ) ^47 , once again we must apply our pattern : 47 = 11*4 +3 , so the third term is equal to 47th term

and we see here the third term is 3 .... so our answer is 3 , so we can say that the unit digit of ( 137^13) ^47 is 3 .... answer B...

Celestial09 · Answer

IMO it should be 1 (A)

this is how I would have done in exam :
power repetitive of 7..
7^1 =7
7^2 = 9
7^3 = 3
7^4 = 1
7^5=7
7^6=9
7^7=3
this means remainders of power ending with 1 gives unit digit 7
remainders of power ending with 2 gives unit digit 9
13/7 gives remainder of 6... hence unit digit left is 9

9^1=9
9^2=1
9^3=9

this means remainders of power ending with 1 gives unit digit 9
remainders of power ending with 2 gives unit digit 1

repetitive nature is 2
47/9 gives remainder 2
hence unit digit is 1

unit digit is 1

Kudos please if my solution is right and having appropriate method
Thanks
Celestial

chetan2u · Answer

hi  Celestial09 ,
your approach is correct but you have gone wrong in your observation of repetitive nature of power of 7.. it is 7,9,3,1,7,9,3,1.. so every term after multiple of 4 starts a new repetition..
this is the standard repetitive nature of powers , they repeat after every 4th term(2,3,7,8).... with some after every 2nd term example 9,4 and few each term is same... eg 1,5,6,0...
so if 7 has 7,9,3,1...13 leaves a remainder of 1 when divided by 4.. so we have xy..7^47..
now 47 leaves a remainder of 3, so last digit is 3..ans B

mvictor · Answer

(137^13)^47
we can rewrite this as 137^(13*47) = 13^611
now let's look for the repetitive pattern
7^1 = units digit 7
7^2 = units digit 9
7^3 = units digit 3
7^4 = units digit 1
etc.

we can see that when raising 7 to a power, which is a multiple of 4, the units digit of the number is 1
we have number 612, which is a multiple of 4, and when 7 raised to 612 power, the units digit is 1. thus we can conclude that for 7 raised to the 611 power, the units digit is 3.

Bunuel · Answer

MAGOOSH OFFICIAL SOLUTION:

First of all, all we need is the last digit of the base, not 137, but just 7. Here’s the power sequence of the units of 7

7^1 has a units digit of 7
7^2 has a units digit of 9 (e.g. 7*7 = 49)
7^3 has a units digit of 3 (e.g. 7*9 = 63)
7^4 has a units digit of 1 (e.g. 7*3 = 21)
7^5 has a units digit of 7
7^6 has a units digit of 9
7^7 has a units digit of 3
7^8 has a units digit of 1
etc.

The period is 4, so 7 to the power of any multiple of 4 has a units digit of 1
7^12 has a units digit of 1
7^13 has a units digit of 7

So the inner parenthesis is a number with a units digit of 7.

Now, for the outer exponent, we are following the same pattern — starting with a units digit of 7. The period is still 4.
7^44 has a units digit of 1
7^45 has a units digit of 7
7^46 has a units digit of 9
7^47 has a units digit of 3

So the unit digit of the final output is 3.

Answer = B

ScottTargetTestPrep · Answer

Simplifying, we have (137^13)^47 = 137^611. Now recall that the units digit of a number raised to a power is the same as the units digit of that number’s units digit raised to the same power. Therefore, we can just determine the units digit of 7^611.

Recall that the units digit pattern of 7 raised to a power is 7-9-3-1. We see that 7 raised to a power that is a multiple of 4 will result in a units digit of 1. Since 611/4 = 152 R 3, we see that the units digit of 7^611 (and hence 137^611) is 3 (i.e., the 3rd number in the units digit pattern).

Answer: B

Abhishek009 · Answer

7^4  Will have units digit  1

So,  7^{13} = 7^{4*3 + 1}  , Thus units digit will be 7

Again  7^{47} = 7^{4*11 + 3}

7^3  Will have units digit  3

Hence, units digit of (137^13)^47 is (B) 3

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The units digit of (137^13)^47 is:

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