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Which of the following numbers is prime?

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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. \(5^{2881}+7^{2881}\)
[Reveal] Spoiler: OA

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Which of the following numbers is prime? [#permalink]

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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.

Answer: A.

Hope it's clear.
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Re: Which of the following numbers is prime? [#permalink]

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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.
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Re: Which of the following numbers is prime? [#permalink]

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New post 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.

Answer: A.

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

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Re: Which of the following numbers is prime? [#permalink]

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New post 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.

Answer: A.

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.
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Re: Which of the following numbers is prime? [#permalink]

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New post 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.
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Re: Which of the following numbers is prime? [#permalink]

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New post 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,
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Re: Which of the following numbers is prime? [#permalink]

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Re: Which of the following numbers is prime? [#permalink]

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New post 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.

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Re: Which of the following numbers is prime? [#permalink]

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New post 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!

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Re: Which of the following numbers is prime? [#permalink]

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New post 16 Apr 2016, 22:41
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
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Re: Which of the following numbers is prime? [#permalink]

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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
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Re: Which of the following numbers is prime? [#permalink]

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New post 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?

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Re: Which of the following numbers is prime? [#permalink]

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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.
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Re: Which of the following numbers is prime? [#permalink]

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New post 12 Jul 2016, 03:29
Bunuel Ah! I missed it totally. Thanks!

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Re: Which of the following numbers is prime? [#permalink]

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New post 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:
Image (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, Image this eliminates C
D - difference can also be easily eliminated.

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Re: Which of the following numbers is prime? [#permalink]

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New post 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

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Re: Which of the following numbers is prime? [#permalink]

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New post 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

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Re: Which of the following numbers is prime? [#permalink]

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New post 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.
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Re: Which of the following numbers is prime?   [#permalink] 18 Dec 2017, 19:54

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Which of the following numbers is prime?

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