Number reference
167 explained.
167: one hundred sixty-seven, prime; factors, prime factorization, roots, binary, hexadecimal, and digit properties.
Identity and representations
- Written form
- one hundred sixty-seven
- Classification
- prime
- Factors
- 1, 167
- Prime factorization
- 167
- Square root
- 12.922848
- Cube root
- 5.506878
- Binary
- 10100111
- Octal
- 247
- Hexadecimal
- A7
- Roman numeral
- CLXVII
- Digit sum
- 14
- Digital root
- 5
What makes 167 notable
- Prime. 167 is prime: it has exactly two divisors, 1 and itself, so it cannot be split into equal whole-number groups any other way.
167 in everyday use
One hundred sixty-seven is a safe prime: subtract one and halve, and the result, 83, is prime as well. Safe primes are valued in cryptography because Diffie–Hellman key exchange and related schemes are harder to attack when the modulus is built from one. It is also a full-reptend prime, meaning the decimal expansion of 1/167 repeats only after the maximum possible 166 digits. Reverse its digits and you get 761, another prime, so 167 is an emirp too.
The number has a footnote in the study of sums of cubes. Every whole number can be written as a sum of at most nine positive cubes, and only 23 and 239 actually need nine; 167 is among the small handful that need eight. Written out, 167 = 125 + 8 + 8 + 8 + 8 + 8 + 1 + 1, using the cubes of 5, 2, and 1. It is the 39th prime, and its digit sum is 14.
Nearby numbers
How to read this page
Because 167 is prime, the factor list stops at 1 and 167 itself, so it cannot be shared into equal whole-number groups. That is also what makes it useful as a building block: every larger number with 167 in its prime factorisation inherits it from here.
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