What is the units digit of 66^6 - 5^55?

Question

A. 0
B. 1
C. 5
D. 6
E. 9

Bunuel · Accepted Answer

Official Solution:

A number with the units digit of 6, in any positive integer power, will have 6 as its units digit. So, the units digit of  66^6  is 6.
 
A number with the units digit of 5, in any positive integer power, will have 5 as its units digit. So, the units digit of  5^{55}  is 5.
 
Here, you have to be careful and not assume that the units digit of  66^6 - 5^{55}  will be  6 - 5 = 1  because you should notice that  5^{55}  is MUCH larger number than  66^6 . Why?  5^{55}=5^{5*11}=(5^5)^{11} . Now,  5^5>66  and  11>6 , thus  (5^5)^{11}>66^6 .
 
Therefore, the units digit of  66^6 - 5^{55}  will be  \{ smaller \ number \ with \ the \ units \ digit \ of \ 6 \} - \{ larger \ number \ with \ the \ units \ digit \ of \ 5\} = 9 . For example,  16-25=-9

Answer: E

NandishSS · Answer

The cycle of 5 and 6, when raised to any power, will be same respectively.

For ex 5^1=5

5^2=25

5^3=125

And 6^1=6

6^2=36

6^3=216

However 66^6 < 5^55 unit digit would be negative

So 6-25 =-9

Ans :  E

shashankism · Answer

Since 6^2 = 36, 6^3 = 216 .. So unit's digit of 66^6 will be 6 
Also 5^2 = 25. 5^3 = 125 , 5^4 = 625 .. So unit's digit of 5^55 will be 5

So, units digit of 66^6 - 5^55 = 6-5 =1

Answer B

Mahmud6 · Answer

This problem is dealing with the last digit of a power. Details are described in the following link https://gmatclub.com/forum/math-number- ... 88376.html.

Integer ending with 0, 1, 5 or 6, in the integer power k>0, has the same last digit as the base. So, the last digit of \(66^6\) is 6 and the last digit of \(5^{55}\) is 5. Therefore, the answer would be xxxx6-xxxx5=1, B.

Bunuel · Answer

Simple tricks are the best... The answer is  not B

mj12345 · Answer

The last digit will be 1 if 66^6 > 5^55.

I don’t know the best way to explain this; but looking at the numbers 5^55 >>> 66^6. So in this case:

xxx6 - xxxxxxx5 = -yyyyy9

The units digit will be 9.

Sent from my iPhone using GMAT Club Forum

NandishSS · Answer

Bunuel Trapped !!! mykbdj Caught It correctly

Agree with his explaination

atulindia · Answer

Unit digit of 66^6 is 6 and 5^55 is 5
But we do not know which one is greater, so unit of difference can be 1 or 9
But from given data 66x66x66x66x66x66 or 66^6 <<< 125x125x125x125x125x125 x 5^37 or 5^55
Hence unit digit is 9 and answer is E

aashaybaindurgmat · Answer

Answer : 9 
Answer : E

Obviously 66^6 will give a units digit of 6.
and 5^55 will give a units digit of 5.

But 6-5 = 1 will only apply if there is no carrying forward meaning, that the "6" number is larger than the "5" Number.

Doing some math though tells us that the "5" number is indeed much larger, which means carrying forward does take place.
Making the subtraction look much more like 15 - 6 = 9, than the aforementioned 6 - 5 = 1.

Therefore the answer is E.

Bunuel · Answer

Par of  GMAT CLUB'S New Year's Quantitative Challenge Set

BrushMyQuant · Answer

We need to find the units digit of 66^6 - 5^55?

Both 6 and 5 have a cycle of 1 for units digit

Watch these videos  to  Master Cyclicity for Unit's Digit

That means any positive power of 5 and 6 will have units digit of 5 and 6 respectively

Example:
Units digit of  5^1  = 5
Units digit of  5^2  = 5
Units digit of  5^3  = 5

Units digit of  6^1  = 6
Units digit of  6^2  = 6
Units digit of  6^3  = 6

But  5^{55} > 6^6   
=> Units digit of 6 6^6 - 5^{55}  = -X5 + 6 = -9 (Similar to 6 - 15)

So,  Answer will be E 
Hope it helps!

Vikramaditya00 · Answer

Excellent question, looking at the question didnt check nor thought if checking which number will be bigger directly calculated and selected 1

anushree01 · Answer

Obviously 66^6 will give a units digit of 6.
and 5^55 will give a units digit of 5.

But 6-5 = 1 will only apply if there is no carrying forward meaning

What is the units digit of 66^6 - 5^55?

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GMAT Problem Solving (PS) Questions

What is the units digit of 66^6 - 5^55?

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