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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