Problem Solving

ardz24 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

OA: A

Unless I'm missing something super obvious, this question requires too many calculations to be GMAT-worthy.
There's no quick way to show that A is prime. Instead, we have to show that B, C, D and E are not prime.

B) Focus on the UNITS digit of each term.
The units digit of 2^31 is 8, and the units digit of 3^31 is 7
So, the units digit of 2^31+3^31 is 5, which means 2^31+3^31 is divisible by 5
So, B is NOT prime

ASIDE: here's an article on finding the units digits of large powers: https://www.gmatprepnow.com/articles/un ... big-powers

C) Focus on the UNITS digit of each term.
The units digit of 4^66 is 6, and the units digit of 7^66 is 9
So, the units digit of 4^66 + 7^66 is 5, which means 4^66 + 7^66 is divisible by 5
So, C is NOT prime

D) 5^82−2^82 is a difference of squares, so it can be factored as follows:
5^82 − 2^82 = (5^41 + 2^41)(5^41 - 2^41)
So, D is NOT prime

E) 5^2881 + 7^2881
5^2881 is ODD and 7^2881 is ODD
So, 5^2881 + 7^2881 = ODD + ODD = EVEN
So, E is NOT prime

By the process of elimination, the correct answer is A

Cheers,
Brent

ardz24 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

OA: A

Is there a strategic approach to this question?

Let's first get the divisibility cycles and deduce the unit digits.

1. 2: Unit digits: (2, 4, 8, 6); the order of unit digits has a cycle of 4.
2. 3: Unit digits: (3, 9, 7, 1); the order of unit digits has a cycle of 4.
3. 4: Unit digits: (4, 6); the order of unit digits has a cycle of 2.
4. 5: Unit digits: (5); unit digit is always 5.

Let's take each option one by one.

A. 2^16+1:

'2' has a cycle of 4, and the exponent 16 is divisible by 4, thus, the unit digit of 2^16 is '6'. Thus, the unit digit of 2^16 + 1 is 6 + 1 = 7. It can be a prime number. We cannot be sure though.

B. 2^31+3^31:

'2' has a cycle of 4; the exponent 31 divided by 4 leaves a remainder 3, thus, the unit digit of 2^31 is '8'.

Similarly, '3' has a cycle of 4; the exponent 31 divided by 4 leaves a remainder 3, thus, the unit digit of 3^31 is '7'.

Thus, the unit digit of 2^31 + 3^31 is 8 + 7 = 15 => Unit digit = 5. Since every number with unit digit 5 is divisible by 5, it's not a prime number.

C. 4^66+7^66

'4' has a cycle of 2; the exponent 66 divided by 4 leaves a remainder 2, thus, the unit digit of 4^66 is '6'.

Similarly, '7' has a cycle of 4; the exponent 66 divided by 4 leaves a remainder 2, thus, the unit digit of 7^66 is '9'.

Thus, the unit digit of 4^66 + 7^66 is 8 + 9 = 15 => Unit digit = 5. Since every number with unit digit 5 is divisible by 5, it's not a prime number.

D. 5^82−2^82

Since the unit digit of 5 raised to any exponent is 5, the unit digit of 5^82 is 5.

'2' has a cycle of 4; the exponent 82 divided by 4 leaves a remainder 2, thus, the unit digit of 2^82 is '4'.

Thus, the unit digit of 5^82 − 2^82 is 5 - 4 = 1. It can be a prime number. We cannot be sure though.

So, as of now, we have two options A and D to choose.

Relook at 5^82 − 2^82

5^82 − 2^82 = (5^41)^2 − (2^41)^2 = [5^41 + 2^41]*[5^41 − 2^41]

Since 5^82 − 2^82 has two factors, it's not prime.

E. 5^2881 + 7^2881

Since 5^2881, and 7^2881 are odd numbers, their sum is even. We know that no number greater than 2 is prime, thus, 5^2881 + 7^2881 is not a prime number.

The correct answer: A

Hope this helps!

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BTGmoderatorAT wrote: ↑

Mon Sep 18, 2017 5:04 am

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

OA: A

Is there a strategic approach to this question?

We can immediately eliminate choice E since it will be an even number > 2, which can’t be a prime. Also we see that choice D is 5^82 - 2^82 = (5^41 - 2^41)(5^41 + 2^41). Since both factors are greater than 1, it can’t be a prime either.

Now let’s focus on the units digit of the 3 remaining choices (recall the units digit pattern of powers of 2 is 2-4-8-6, that of 3 is 3-9-7-1, that of 4 is 4-6, and that of 7 is 7-9-3-1).

A. 2^16 + 1: units digit = 6 + 1 = 7

B. 2^31 + 3^31: units digit = 8 + 7 = 15, i.e., 5

C. 4^66 + 7^66: units digit = 6 + 9 = 15, i.e., 5

We see that the units digit of both choices B and C is 5 and since they are obviously greater than 5, they can’t be prime either. Therefore, the correct answer must be A.

Answer: A