The number 8191

8191 is a Mersenne prime (2^13 − 1); the 1028th prime number; a binary palindrome (1111111111111).

8191 is a Mersenne prime ( $2^{13}$ − 1), the 1028th prime number and a binary palindrome (1111111111111). Its only divisors are 1 and 8191.

Number

Pick any whole number from 1 to 100,000.

8191\ \text{is prime}

How to work it out

1
What is special about 8191: a Mersenne prime ( $2^{13}$ − 1); the 1028th prime number; a binary palindrome (1111111111111).
2
8191 is prime, so its only factors are 1 and 8191. It is the 1028th prime; the next prime is 8209 and the one before is 8179.
3
The divisors add up to 8192; without 8191 itself that is 1, so 8191 is deficient (1 is less than 8191). Euler's totient φ(8191) = 8190 numbers up to 8191 share no factor with it.
4
The digits of 8191 add up to 19, and repeating that until one digit is left gives the digital root 1. The Collatz sequence (halve if even, else 3n+1) takes 158 steps to reach 1.

Form	Value
In words	eight thousand one hundred and ninety-one
Binary (base 2)	1111111111111
Octal (base 8)	17777
Hexadecimal (base 16)	1FFF
Base 3	102020101
As time	8191 seconds = 2 h 16 min 31 s

Divisor	Answer	Rule
2	No (remainder 1)	A number is divisible by 2 if the last digit is even (0, 2, 4, 6, 8).
3	No (remainder 1)	A number is divisible by 3 if the digits add up to a multiple of 3.
4	No (remainder 3)	A number is divisible by 4 if the last two digits make a multiple of 4.
5	No (remainder 1)	A number is divisible by 5 if the last digit is 0 or 5.
6	No (remainder 1)	A number is divisible by 6 if it is divisible by both 2 and 3.
7	No (remainder 1)	A number is divisible by 7 if double the last digit and subtract it from the rest: the result is a multiple of 7.
8	No (remainder 7)	A number is divisible by 8 if the last three digits make a multiple of 8.
9	No (remainder 1)	A number is divisible by 9 if the digits add up to a multiple of 9.
10	No (remainder 1)	A number is divisible by 10 if the last digit is 0.
11	No (remainder 7)	A number is divisible by 11 if the alternating sum of the digits is a multiple of 11.
12	No (remainder 7)	A number is divisible by 12 if it is divisible by both 3 and 4.

Is 8191 a prime number?

Yes. 8191 is divisible only by 1 and itself.

What are the factors of 8191?

1, 8191.

What is 8191 in binary?

1111111111111 (hexadecimal 1FFF, octal 17777).

What is the square root of 8191?

About 90.5041.

Is 8191 divisible by 3?

No. Its digits add up to 19, which is not a multiple of 3.

What is special about the number 8191?

8191 is a Mersenne prime ( $2^{13}$ − 1); the 1028th prime number; a binary palindrome (1111111111111).