GMAT Question of the Day - Daily to your Mailbox; hard ones only

 It is currently 19 Feb 2019, 09:07

### GMAT Club Daily Prep

#### Thank you for using the timer - this advanced tool can estimate your performance and suggest more practice questions. We have subscribed you to Daily Prep Questions via email.

Customized
for You

we will pick new questions that match your level based on your Timer History

Track

every week, we’ll send you an estimated GMAT score based on your performance

Practice
Pays

we will pick new questions that match your level based on your Timer History

## Events & Promotions

###### Events & Promotions in February
PrevNext
SuMoTuWeThFrSa
272829303112
3456789
10111213141516
17181920212223
242526272812
Open Detailed Calendar
• ### Free GMAT Prep Hour

February 20, 2019

February 20, 2019

08:00 PM EST

09:00 PM EST

Strategies and techniques for approaching featured GMAT topics. Wednesday, February 20th at 8 PM EST

February 21, 2019

February 21, 2019

10:00 PM PST

11:00 PM PST

Kick off your 2019 GMAT prep with a free 7-day boot camp that includes free online lessons, webinars, and a full GMAT course access. Limited for the first 99 registrants! Feb. 21st until the 27th.

# Which of the following numbers is prime?

Author Message
TAGS:

### Hide Tags

Senior Manager
Status: Verbal Forum Moderator
Joined: 17 Apr 2013
Posts: 477
Location: India
GMAT 1: 710 Q50 V36
GMAT 2: 750 Q51 V41
GMAT 3: 790 Q51 V49
GPA: 3.3
Which of the following numbers is prime?  [#permalink]

### Show Tags

01 Mar 2014, 22:56
24
85
00:00

Difficulty:

25% (medium)

Question Stats:

71% (01:36) correct 29% (01:48) wrong based on 1059 sessions

### HideShow timer Statistics

Which of the following numbers is prime?

A. $$2^{16}+1$$

B. $$2^{31}+3^{31}$$

C. $$4^{66}+7^{66}$$

D. $$5^{82}−2^{82}$$

E. $$5^{2881}+7^{2881}$$

_________________

Like my post Send me a Kudos It is a Good manner.
My Debrief: http://gmatclub.com/forum/how-to-score-750-and-750-i-moved-from-710-to-189016.html

Math Expert
Joined: 02 Sep 2009
Posts: 52971
Which of the following numbers is prime?  [#permalink]

### Show Tags

02 Mar 2014, 04:18
29
18
honchos wrote:
Which of the following numbers is prime?

A. 2^16+1
B. 2^31+3^31
C. 4^66+7^66
D. 5^82−2^82
E. 5^2881+7^2881

Let's check which of the options is NOT a prime:

A. $$2^{16} + 1$$ --> the units digit of 2 in positive integer power repeats in blocks of four {2, 4, 8, 6}. Hence, the units digit of 2^16 is 6 and the units digit of 2^16 + 1 is 7 --> 2^16 + 1 CAN be a prime.

B. $$2^{31} + 3^{31}$$ --> the units digit of 2^31 is 8 and the units digit of 3^31 is 7 (the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}). Hence, the units digit of 2^31 + 3^31 is 5 (8+7). Thus 2^31 + 3^31 is divisible by 5. Not a prime.

C. $$4^{66} + 7^{66}$$ --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6}) and the units digit of 7^66 is 9 (the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}). Hence, the units digit of 4^66 + 7^66 is 5 (6+9). Thus 4^66 + 7^66 is divisible by 5. Not a prime.

D. $$5^{82} - 2^{82}$$ --> we can factor this as (5^41 - 2^41)(5^41 + 2^41). Not a prime.

E. $$5^{2881}+ 7^{2881}$$ --> 5^2881 + 7^2881 = odd + odd = even. Not a prime.

Only option A can be prime.

Hope it's clear.
_________________
##### General Discussion
Senior Manager
Status: Verbal Forum Moderator
Joined: 17 Apr 2013
Posts: 477
Location: India
GMAT 1: 710 Q50 V36
GMAT 2: 750 Q51 V41
GMAT 3: 790 Q51 V49
GPA: 3.3
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

02 Mar 2014, 04:37
3
Thanks for the Explanation Bunuel. This questions has many concepts. infact this question cleared my many questions. So I though to share and post so that others could benefit from it.
_________________

Like my post Send me a Kudos It is a Good manner.
My Debrief: http://gmatclub.com/forum/how-to-score-750-and-750-i-moved-from-710-to-189016.html

Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

02 Mar 2014, 04:39
honchos wrote:
Thanks for the Explanation Bunuel. This questions has many concepts. infact this question cleared my many questions. So I though to share and post so that others could benefit from it.

Thank you for posting.
_________________
Intern
Joined: 18 Feb 2014
Posts: 9
Location: United States
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

02 Mar 2014, 13:24
Bunuel wrote:
honchos wrote:
Which of the following numbers is prime?

A. 2^16+1
B. 2^31+3^31
C. 4^66+7^66
D. 5^82−2^82
E. 5^2881+7^2881

Let's check which of the options is NOT a prime:

A. 2^16 + 1 --> the units digit of 2 in positive integer power repeats in blocks of four {2, 4, 8, 6}. Hence, the units digit of 2^16 is 6 and the units digit of 2^16 + 1 is 7 --> 2^16 + 1 CAN be a prime.

B. 2^31 + 3^31 --> the units digit of 2^31 is 8 and the units digit of 3^31 is 7 (the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}). Hence, the units digit of 2^31 + 3^31 is 5 (8+7). Thus 2^31 + 3^31 is divisible by 5. Not a prime.

C. 4^66 + 7^66 --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6}) and the units digit of 7^66 is 9 (the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}). Hence, the units digit of 4^66 + 7^66 is 5 (6+9). Thus 4^66 + 7^66 is divisible by 5. Not a prime.

D. 5^82 - 2^82 --> we can factor this as (5^41 - 2^41)(5^41 + 2^41). Not a prime.

E. 5^2881+ 7^2881 --> 5^2881 + 7^2881 = odd + odd = even. Not a prime.

Only option A can be prime.

Hope it's clear.

Bunuel,

..... C. 4^66 + 7^66 --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6})..... 66/2 (number of repetitions of two) is 33, and therefore it should be the first number of the block of 2, meaning the units digits is 4. I think I am missing something
Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

02 Mar 2014, 23:41
franloranca wrote:
Bunuel wrote:
honchos wrote:
Which of the following numbers is prime?

A. 2^16+1
B. 2^31+3^31
C. 4^66+7^66
D. 5^82−2^82
E. 5^2881+7^2881

Let's check which of the options is NOT a prime:

A. 2^16 + 1 --> the units digit of 2 in positive integer power repeats in blocks of four {2, 4, 8, 6}. Hence, the units digit of 2^16 is 6 and the units digit of 2^16 + 1 is 7 --> 2^16 + 1 CAN be a prime.

B. 2^31 + 3^31 --> the units digit of 2^31 is 8 and the units digit of 3^31 is 7 (the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}). Hence, the units digit of 2^31 + 3^31 is 5 (8+7). Thus 2^31 + 3^31 is divisible by 5. Not a prime.

C. 4^66 + 7^66 --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6}) and the units digit of 7^66 is 9 (the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}). Hence, the units digit of 4^66 + 7^66 is 5 (6+9). Thus 4^66 + 7^66 is divisible by 5. Not a prime.

D. 5^82 - 2^82 --> we can factor this as (5^41 - 2^41)(5^41 + 2^41). Not a prime.

E. 5^2881+ 7^2881 --> 5^2881 + 7^2881 = odd + odd = even. Not a prime.

Only option A can be prime.

Hope it's clear.

Bunuel,

..... C. 4^66 + 7^66 --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6})..... 66/2 (number of repetitions of two) is 33, and therefore it should be the first number of the block of 2, meaning the units digits is 4. I think I am missing something

Consider the following example: what is the units digit of 127^124.

First of all, the units digit of 127^124 is the same as that of 7^124 (get rid of all the digits except the units digit).

Next, recall that the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}.

Finally, to get the units digit of 7^124, you need to divide the exponent (124) by 4 (cyclicity) and look at the remainder you get:

Remainder = 1 --> the units digit = 1st number from the pattern, so 7.
Remainder = 2 --> the units digit = 2nd number from the pattern, so 9.
Remainder = 3 --> the units digit = 3rd number from the pattern, so 3.
Remainder = 0 --> the units digit = 4th number from the pattern, so 1.

Now, since 124/4 yields the remainder of 0 (124 is divisible by 4), then the units digit of 7^124 is 1.

We can apply the same logic to 4^66: the units digit of 4 in positive integer power repeats in blocks of two {4, 6} --> 66/2 yields the remainder of 0, thus the units do digit of 4^66 is 2nd number from the pattern, so 6. Or another way: 4^odd has the units digit of 4 and 4^even has the units digit of 6.

For more check Number Theory chapter of our Math Book: math-number-theory-88376.html

Hope it's clear.
_________________
Retired Moderator
Joined: 20 Dec 2013
Posts: 171
Location: United States (NY)
GMAT 1: 640 Q44 V34
GMAT 2: 710 Q48 V40
GMAT 3: 720 Q49 V40
GPA: 3.16
WE: Consulting (Venture Capital)
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

19 Mar 2014, 18:59
I narrowed it down to A & D as they end in a 7 and 1 respectively (the rest end in 5 or are even).

D is equivalent to difference of squares (5^41)^2 - (2^41)^2 with each 5 and 2 to the equivalent of their 1st power, so 5^2-2^2 = 21, then I tried an equivalent of 4th power of 2 + 1, or 2^4 + 1 = 17, so I picked A.
_________________
Manager
Joined: 30 May 2013
Posts: 153
Location: India
Concentration: Entrepreneurship, General Management
GPA: 3.82
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

09 Jul 2014, 07:55
Hi Bunnel,,
Can we take as

a^(Square of any number) + b^ (square of any number) = prime number????

Clear my doubt.

Regards,
RRsnathan.
Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

09 Jul 2014, 07:58
rrsnathan wrote:
Hi Bunnel,,
Can we take as

a^(Square of any number) + b^ (square of any number) = prime number????

Clear my doubt.

Regards,
RRsnathan.

What do you mean exactly???

2^16+1 = 65,537, which IS a prime number.
_________________
Manager
Joined: 30 May 2013
Posts: 153
Location: India
Concentration: Entrepreneurship, General Management
GPA: 3.82
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

10 Jul 2014, 05:22
Bunuel wrote:
rrsnathan wrote:
Hi Bunnel,,
Can we take as

a^(Square of any number) + b^ (square of any number) = prime number????

Clear my doubt.

Regards,
RRsnathan.

What do you mean exactly???

2^16+1 = 65,537, which IS a prime number.

2^16+1^16 = Prime
Yes Bunnel. Even i tried with couple of combination. I got prime numbers.
Intern
Joined: 29 May 2015
Posts: 4
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

27 Jan 2016, 19:58
Bunuel

Could you clarify two things -
If we were to use the same approach that we used for answer A, B and C, and apply them for answer E, we get:

Units digit for option E is 2 (U5 + U7) = U12, and thus this is not prime.
Is this deduction correct? I'm trying to understand whether we can use the same approach for E. Did you use your approach for time saving purposes?

Additionally, can you help me understand the concept behind the answer you obtained for D
5^82 - 2^82 --> we can factor this as (5^41 - 2^41)(5^41 + 2^41). Not a prime.
I don't fully understand why the factorization helps us deduce that this number is not prime.

Thank you as always!
Intern
Joined: 08 Feb 2016
Posts: 37
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

16 Apr 2016, 22:41
1
Additionally, can you help me understand the concept behind the answer you obtained for D
5^82 - 2^82 --> we can factor this as (5^41 - 2^41)(5^41 + 2^41). Not a prime.
I don't fully understand why the factorization helps us deduce that this number is not prime.

Let us say "5^82 - 2^82" results in some number x. If we are able to factor x as x=a*b (a and b are two numbers, can be same or different), this means that x is not a prime. Since any prime number will have 1 and itself as the ONLY factors.

Thanks
_________________

Once you know the answer, it is easy to justify.

Current Student
Joined: 12 Aug 2015
Posts: 2621
Schools: Boston U '20 (M)
GRE 1: Q169 V154
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

22 Apr 2016, 13:22
1
The Rule to be used here is that the primes >5 cannot have UD as 0,2,4,5,6,8 hence they must end with 1 or 3 or 7 or 9
Fortunately A satisfies That
_________________
Current Student
Joined: 28 Nov 2014
Posts: 846
Concentration: Strategy
Schools: Fisher '19 (M$) GPA: 3.71 Re: Which of the following numbers is prime? [#permalink] ### Show Tags 11 Jul 2016, 23:10 Bunuel wrote: D. 5^82 - 2^82 --> we can factor this as (5^41 - 2^41)(5^41 + 2^41). Not a prime. Hope it's clear. Bunuel I would like to understand how if an option can be written as a factor (a-b)(a+b) cannot be Prime? Math Expert Joined: 02 Sep 2009 Posts: 52971 Re: Which of the following numbers is prime? [#permalink] ### Show Tags 12 Jul 2016, 02:02 Keats wrote: Bunuel wrote: D. 5^82 - 2^82 --> we can factor this as (5^41 - 2^41)(5^41 + 2^41). Not a prime. Hope it's clear. Bunuel I would like to understand how if an option can be written as a factor (a-b)(a+b) cannot be Prime? A prime number has only two factors: 1 and itself. We broke 5^82 - 2^82 into the product of two factors different from 1 and 5^82 - 2^82 itself, so 5^82 - 2^82 is not a prime. _________________ Current Student Joined: 28 Nov 2014 Posts: 846 Concentration: Strategy Schools: Fisher '19 (M$)
GPA: 3.71
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

12 Jul 2016, 03:29
Bunuel Ah! I missed it totally. Thanks!
Manager
Joined: 20 Sep 2016
Posts: 108
GMAT 1: 680 Q49 V35
GPA: 3.99
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

13 Apr 2017, 13:24
Bunuel, can we use here the following property?

If n is odd, we can factor $$a^n + b^n$$ like this:
(wikipedia)
B and E can be ruled out if the factor after (a+b) is not equal to one. I guess it cannot be since 2^31+3^31 = (2+3)*(sth. greater than one)

Additionaly, If n is even, we consider two cases (again from wiki):
If n is a power of 2 then $$a^n + b^n$$ is unfactorable (more precisely, irreducible over the rational numbers). -this alone gives us the answer
Otherwise, this eliminates C
D - difference can also be easily eliminated.
Intern
Joined: 18 Jul 2017
Posts: 5
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

17 Sep 2017, 09:43
Remainder = 1 --> the units digit = 1st number from the pattern, so 7.
Remainder = 2 --> the units digit = 2nd number from the pattern, so 9.
Remainder = 3 --> the units digit = 3rd number from the pattern, so 3.
Remainder = 0 --> the units digit = 4th number from the pattern, so 1.
how do u get the remainder 1,2,3,0.
plz help
Manager
Joined: 17 Sep 2017
Posts: 55
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

18 Dec 2017, 18:35
Quote:
B. 231+331231+331 --> the units digit of 2^31 is 8 and the units digit of 3^31 is 7 (the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}). Hence, the units digit of 2^31 + 3^31 is 5 (8+7). Thus 2^31 + 3^31 is divisible by 5. Not a prime.

C. 466+766466+766 --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6}) and the units digit of 7^66 is 9 (the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}). Hence, the units digit of 4^66 + 7^66 is 5 (6+9). Thus 4^66 + 7^66 is divisible by 5. Not a prime.

I don't understand how do we get 5 here. Is there any formula that I don't know? Need help
Math Expert
Joined: 02 Sep 2009
Posts: 52971
Re: Which of the following numbers is prime?  [#permalink]

### Show Tags

18 Dec 2017, 19:54
lichting wrote:
Quote:
B. 231+331231+331 --> the units digit of 2^31 is 8 and the units digit of 3^31 is 7 (the units digit of 3 in positive integer power repeats in blocks of four {3, 9, 7, 1}). Hence, the units digit of 2^31 + 3^31 is 5 (8+7). Thus 2^31 + 3^31 is divisible by 5. Not a prime.

C. 466+766466+766 --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6}) and the units digit of 7^66 is 9 (the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}). Hence, the units digit of 4^66 + 7^66 is 5 (6+9). Thus 4^66 + 7^66 is divisible by 5. Not a prime.

I don't understand how do we get 5 here. Is there any formula that I don't know? Need help

$$4^{66} + 7^{66}$$ --> the units digit of 4^66 is 6 (the units digit of 4 in positive integer power repeats in blocks of two {4, 6}) and the units digit of 7^66 is 9 (the units digit of 7 in positive integer power repeats in blocks of four {7, 9, 3, 1}). Hence, the units digit of 4^66 + 7^66 is 5 (6+9). Thus 4^66 + 7^66 is divisible by 5. Not a prime.

The units digit of 4^(positive integer) repeats in blocks of two - {4, 6}:
4^1 = 4;
4^2 = 16;
4^3 = 8\frac{4[}{fraction];
4^4 = 256;
...

So, the inits digit of 4^256 will be 6 (the odd powers give 4 and even powers give 6).

The units digit of 7^(positive integer) repeats in blocks of four - {7, 9, 3, 1}:
7^1 = 7;
7^2 = 49;
7^3 = ...[fraction]3};
7^4 = ...1;
7^5 = ...7 (7 again)
...

So, the inits digit of 7^66 will be 9. Divide 66 (power) by 4 (cyclisity), remainder is 2. So, the units digit of 7^66 is the same as that of the units digit of 7^2, which is 9.

Hence, the units digit of 4^66 + 7^66 is 5 (6+9).

Theory is here: https://gmatclub.com/forum/math-number- ... 88376.html

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

Hope it helps.
_________________
Re: Which of the following numbers is prime?   [#permalink] 18 Dec 2017, 19:54

Go to page    1   2    Next  [ 24 posts ]

Display posts from previous: Sort by