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Jitesh8
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Which of the following numbers is prime? 03 Dec 2022, 00:22
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This is really a nice question.
Every GMAT aspirants must attempt this question.

A- 2^16 + 1
May or May not be prime.
B-2^31 +3^31
2 & 3 both Repeats after every 4th power so 8+7=15,always divisible by 5 Hence Non prime no.
C-4^66+7^66
4 Repeats after every 2nd power and 7 Repeats after every 4th power, so 6+9=15, always divisible by 5 Hence Non prime no.
D-5^82-2^82
5^41-2^41(5^41+2^41)
Since it can be factorised into at-least two no. if further factorization is not possible. Hence Non prime no.
E-5^2881+7^2881
5+7=12 Non Prime no. as any even no can always be divided by 2.

NOTE TO SELF.
1-Get rid of all the digits except the units digit and use cyclicity of numbers.
2-use (a^2-b^2)= (a-b)(a+b).
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Which of the following numbers is prime? 23 Dec 2024, 06:40
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how do you find cyclicity of 3^31
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Bunuel
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Which of the following numbers is prime? 23 Dec 2024, 06:44
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onlymalapink
how do you find cyclicity of 3^31
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It explained on the previous page. You ca also check LAST DIGIT OF A POWER chapter here: https://gmatclub.com/forum/math-number- ... 88376.html
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Which of the following numbers is prime? 11 May 2025, 11:58
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A very important property to remember while solving this problem is that

All prime numbers end with 1, 3, 7 or 9 except the prime numbers 2 and 5.
If you know this, option A itself will close your hunt for the answer. You even don't have to evaluate the rest of the answers.
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Which of the following numbers is prime? 28 Jul 2025, 03:42
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Bunuel: Is there any approach like "Only prime numbers have even number of factors"? 2^16 has 17 factors + 1 factor = 18 factors (even)...so it ́s prime. Is this a wrong way to do this question?

Thanks
Bunuel
honchos
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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Which of the following numbers is prime? 28 Jul 2025, 07:12
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RicharddelPino
Bunuel: Is there any approach like "Only prime numbers have even number of factors"? 2^16 has 17 factors + 1 factor = 18 factors (even)...so it ́s prime. Is this a wrong way to do this question?

Thanks
Bunuel
honchos
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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Prime numbers have exactly two positive factors: 1 and the number itself. Perfect squares have an odd number of positive factors. For example, 4 = 2^2 has three factors: 1, 2, and 4. The reverse is also true: if an integer has an odd number of factors, it must be a perfect square.

However, integers that are not perfect squares have an even number of positive factors. For example, 6 is not a perfect square and has four factors: 1, 2, 3, and 6. So, just knowing that a number has an even number of factors does not mean it is prime. A number is prime only if it has exactly two positive factors.

Now, 2^16 is a perfect square and has 17 positive factors. But that does not mean 2^16 + 1 has 18 factors. For example, 2^3 = 8 has four factors, but 2^3 + 1 = 9 has only three: 1, 3, and 9. Even if we knew that 2^16 + 1 had an even number of factors, that alone would not imply it is prime. It could simply be a non-square composite.

Hope it helps.
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Which of the following numbers is prime? 29 Sep 2025, 23:41
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This is a beautiful question because it has more than one way of arriving at the same answer. One way that I came up with immediately that closely parallels some of the other approaches shared in this thread goes as follows:

We know that \((a^n - b^n)\) is divisible by \((a-b)\) for all \(n\). Hence, option D - 5^82 - 2^82 is divisible by \(5-2=3\). Hence, it can be eliminated as it is composite.

Also,

We know that \((a^n + b^n)\) is divisible by \(a+b\) for odd \(n\).

Therefore, Option B: 2^31 + 1^31 is divisible by \(2+3 = 5\). Hence, composite.

Similarly, Option E: 5^2881 + 7^2881 is divisible by \(5+7 = 12\).

Option C isn't as straightforward initially, but if we rewrite it as: (4^2)^33 + (7^2)^33, it is divisible by \(4^2 + 7^2 = 65\).

Therefore, we are left with only A that can be prime, after eliminating other options.
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