If a^b = c, what is the units digit of c?

Question

Competition Mode Question

(1) a^2 – 36 = 0
(2) b is a positive integer

eakabuah · Answer

Given a^b=c, we are to determine the unit digits of c.

Statement 1: a^2 - 36=0
This implies a=-6 or a=6
No information is available on the values of b or c. Hence we cannot determine the unit digit of c. Much as it is tempting to assume that b=2 and c=36, such an assumption is not erroneous. if b=0, then c=1, and its unit digit is 1. However when b=1, c=6 with a unit digit of 6. Statement 1 is insufficient.

Statement 2: b is a positive integer.
Not sufficient. b can take infinite values. In addition, no information is provided on the possible value of a to enable us to determine the value of c leading to the determination of the unit digits of c. Statement 2 is insufficient.

1+2
a=-6 or 6 and b is a positive integer. 
both statements taken together is sufficient. This is because when b=1, the unit digit of c(6)=6. When b=2, the unit digit of c(36) = 6. The unit digit of the positive integer powers of 6 is 6.

The answer is therefore C .

unraveled · Answer

If  a^b = c , what is the units digit of c?

(1)  a^2 – 36 = 0 
 a^2 – 6^2 = 0 
 (a - 6) * (a + 6) = 0

a = 6 or -6. Since either of 6 or -6 raised to any power would always give unit digit as 6, unit digit of c is always 6.

SUFFICIENT.

(2) b is a positive integer

Nothing concrete about both a and b is given so.

INSUFFICIENT.

IMO Answer A.

madgmat2019 · Answer

(1) a^2 – 36 = 0
Then a^2=36...I.e a=6....
If b is positive...
Units digit of a^b will be 6 always
If b is 0
Units digit of a^b will be 1
If b is negative 
Units digit of a^b will be different
So A is insufficient

(2) b is a positive integer
Clearly insuff...Without the value of a

CombiNing both will get units digit as 6

Posted from my mobile device

LalitaSiri · Answer

If a^b = c, what is the units digit of c?

(1) a^2 – 36 = 0
a = 6, -6 
but we do not know anything about b. If b =1, unit of c (if a =6, a = -6) will be 6
However, if b = 0, unit of c = 1
INSUFFICIENT

(2) b is a positive integer
we do not know anything about a. - insufficient

taken (1) + (2): 
we now know that b is always positive
so either a = 6, -6
unit of c will be 6.

Therefore, C is the correct answer

Kinshook · Answer

If a^b = c, what is the units digit of c?

(1) a^2 – 36 = 0
a = {6,-6}
Since value of b is unknown
NOT SUFFICIENT

(2) b is a positive integer
Since value of a is unknown
NOT SUFFICIENT

(1) + (2)
(1) a^2 – 36 = 0
a = {6,-6}
(2) b is a positive integer
Unit digit of c will be 6
SUFFICIENT

IMO C

CareerGeek · Answer

(1) a^2 – 36 = 0
--> a = ±6
But we do not know whether b is an integer or a fraction
So, we cannot have a definite unit digit -->  Insufficient

(2) b is a positive integer
Does not talk anything about ' a ' -->  Insufficient

Combining (1) & (2),
Since Cyclicity of 6 is 1 and b is a positive integer
--> a^b will have end in 6
--> Unit digit of a^b = c is 6 -->  Sufficient

IMO Option C

Archit3110 · Answer

#1
a^2 – 36 = 0
a=+/-6
so any power of b except 0 will give unit digit of c as 6; but b is not know it can be 0 at which c =1
insufficient
#2 b is a positive integer
insufficent
a not know
from 1 &2
b>0 and a = +/-6 ; unit digit of c = 6 always
IMO C

If a^b = c, what is the units digit of c?

(1) a^2 – 36 = 0
(2) b is a positive integer

exc4libur · Answer

(1) a^2 – 36 = 0   insufic. 
 a^2 – 36 = 0…a^2=36…|a|=(6,-6) 
 b=0:a^0 = c…units=1 
 b≠0:(6,-6)^b = c…units=6

(2) b is a positive integer   insufic.

(1 & 2)   sufic. 
 b≠0,a=(6,-6):(6,-6)^{≠0} = c…units=6

Answer (C)

lacktutor · Answer

a^{b} = c—> what is the units digit of c?

(Statement1): a^{2}—36=0
(a—6)(a+6)= 0
a= 6, a=—6 
But no info about what b is.
—> if b=0, then 6^{0}=1 and (—6)^{0}= 1 —> units digit of c is equal to 1

—> if b = 2, then 6^{2}=36 and (—6)^{2}= 36 —> units digit of c is equal to 6

Insufficient

(Statement2) b is a positive integer. 
No info about what a is.
Clearly insufficient.

Taken together 1&2,
—> 6 or (—6) to the power of any positive integer end with ...6 .

Units digit of c is equal to 6 
Sufficient

The answer is C

Posted from my mobile device

siddharthkapoor · Answer

(1)  a^2 =36
=>a=+6 or -6
From this we don't get a unique value of unit digit of  a^b  because for a=6 and b=2,  a^b = 2^6 =36, but for a=6 and b=1/2,  a^b=6^{1/2} =2.45
Therefore, insufficient.

(2) No information about value of a, so we don't get a unique value of  a^b .
Therefore, insufficient.

From (1) and (2), we get that a=+6 or -6 and that b is a positive integer.
So, for all the above values of a and b,  a^b  will have 6 as unit digit, for example,  6^2=36, -6^3=-216 , and so on.
Thus, sufficient.

Therefore, the correct answer is option C.

chaitralirr · Answer

If a^b = c, what is the units digit of c?

(1) a^2 – 36 = 0
(2) b is a positive integer

Statement 1: a^2=36, a=+6 or -6. if a^2=36 units digit is 6.

If a^3= units digit is 2. Hence cannot determine.

Statement 2: b is positive. It could be any number 1 to 3 hence we cannot determine.

when combining statements 1 and 2 still even if b is positive we cannot determine exactly the units digit.

IMO E

PP777 · Answer

Given: a^b= c
 a) a^2 - 36= 0
 => a= 6, a= -6

No info about b. so, Insufficient

b) b is a positive integer.
 no info about a. Insufficient

a) + b) => cyclicity of 6 is 1 and b is positive integer. 
so, unit digit of c is 6.

Correct answer: C

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

If a^b = c, what is the units digit of c?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards