If r, s, and t are all positive integers, what is the remainder of 2^p

Question

Tough and Tricky questions: Remainders.

If r, s, and t are all positive integers, what is the remainder of 2^p/10, if p = rst?

(1) s is even 
(2) p = 4t

Bunuel · Answer

Check  Units digits, exponents, remainders problems directory  in our  Special Questions Directory .

WoundedTiger · Answer

Sol: The questions becomes what is the remainder when 2^(rst)/10 ?

St 1 says: s is even that means r*s*t=even

Now 2^(Even number) can end is 4 or 6..So we have to remainders possible. A and D ruled out

St 2 says p=4*t or p is a multiple of 4 so we will have expression of the form 2^4 or 2^8 or 2^12...all end up in 6..

So remainder will be 6..

Ans is B

Nunuboy1994 · Answer

Bunuel  how do we know whether or not the question reads 2^(p/10) pr (2^p) /10?

Bunuel · Answer

2^p/10 can mean nothing but (2^p)/10. If it were 2^(p/10), it would have been written that way.

gmatway · Answer

Hi,

I think I have missed some concept . can you help me with how 2^4 or 2^8 or 2^12...all end up in 6..?

lacktutor · Answer

p=rst

Statement2: p=4t
—>  \frac{2^p}{10} = \frac{2^{4t}}{10} =  \frac{16^t}{10} =  \frac{(10+6)^t}{10}

—>  \frac{(10^t+10^{t-1}*6^1+...+6^t)}{10}

Each term is divided by 10, except the last one,  6^t.

If you divide  6^t  by 10, 
—> no matter what kind of positive integer t is, the last digit of it will be 6. 
The remainder will be always 6. 
Sufficient

The answer choice is B.

gmatway · Answer

I also observed that the unit digit of the numerator (cyclicity of numerator) is deciding the Remainder when we divide the numerator by 10. 
FOR EG 2^even no. would give us 4/10 or 16/10 or 64/10 so REMAINDER will be 4,6,4 and so on . i.e unit digits .

In option B 2^4t /10 = 16^t/10 and 6 being the unit digit has only 6 in cycle and hence the remainder will be 6 always .

dabaobao · Answer

Concept Tested  
Cyclicity

2 has a cyclicity of 4: 2^1 => 2, 2^2 => 4, 2^3 => 8, 2^4 => 6 (Focus on the unit digit)

1) Remainder could be 4 or 6 => Not sufficient
2) Remainder would always be 6. => Sufficient => B

tamalmallick · Answer

Any number when divided by 10, remainder will be unit's digit. For example, 51 divided by 10 has a
remainder of 1. This question asks for the remainder when an
integer power of 2 is divided by 10. Powers of 2
(2, 4, 8, 16, 32, 64…), we see that the units digit
alternates in a consecutive pattern of 2, 4, 8, 6. Thus, we need to know which of the above mentioned possible units digits we
have with 2p.

(1) INSUFFICIENT: If s is even, we know that the product rst is
even and so is p. Then units digit will be either 4 or 6 (22 = 4, 24 = 16....).

(2) SUFFICIENT: If p = 4t and t is an integer, p must be a multiple
of 4. Since every fourth power of 2 ends with 6 (4 = 16, 8
= 256...), we know that the remainder when 2p is divided by 10
is 6.
The correct answer is B.

Kudos if you like the explanation.

Basshead · Answer

If r, s, and t are all positive integers, what is the remainder of  \frac{2^{p}}{10}, if p = rst ?

(1) s is even

This statement only tells us that rst = even. We can get multiple values of rst, resulting in multiple values of 2^{rst}. Insufficient.

(2) p = 4t

If p = 4t, we know that p = multiple of 4.

The cyclicity of 2 = 2, 4, 8, 6.

Knowing that p = a multiple of four means that the 2^p ALWAYS has a units digit of 6.

6 / 10 = remainder 6. Statement 2 is sufficient.

Answer is B.

aritranandy8 · Answer

VeritasKarishma  plz explain

KarishmaB · Answer

Remainder of 2^p/10 will be decided by the units digit of 2^p.
Units digit of 2^p will be decided by where p appears in the cycle of multiples of 4.

(1) s is even

If s is even, p is even. But do we know whether it is of the form 4a or 4a+2? No. 2^{4a} has units digit of 6 and 2^{4a+2} has units digit of 4. Not sufficient.

(2) p = 4t

p is a multiple of 4 which means that it is of the form 4a. So 2^p will have units digit of 6. So remainder when 2^p is divided by 10 will be 6.
Sufficient

Answer (B)

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 r, s, and t are all positive integers, what is the remainder of 2^p

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