693 (six hundred
ndninety-three) is the
natural number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country").
Numbers used for counting are called ''cardinal ...
following
692
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Year 692 (Roman numerals, DCXCII) was a leap year starting on Monday (link will display the full calendar) of the Julian calendar. The denomination 692 for this year has been used since the early medieval period, when the Anno Domini ...
and preceding
694.
In mathematics
693 has twelve divisors: 1, 3, 7, 9, 11, 21, 33, 63, 77, 99, 231, and 693. Thus, 693 is tied with
315
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Year 315 ( CCCXV) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Constantinus and Licinianus (or, less frequently, year ...
for the highest number of divisors for any odd natural number below 900. The smallest positive odd integer with more divisors is 945, which has 16 divisors. Consequently, 945 is also the smallest odd
abundant number
In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. Th ...
, having an abundancy index of 1920/945 ≈ 2.03175.
693 appears as the first three digits after the decimal point in the decimal form for the
natural logarithm of 2. To 10 digits, this number is 0.6931471805. As a result, if an event has a constant
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
of 0.1% of occurring, 693 is the smallest number of trials that must be performed for there to be at least a 50% chance that the event occurs at least once. More generally, for any probability p, the probability that the event occurs at least once in a sample of n items, assuming the items are independent, is given by the following formula:
1 − (1 − p)
n
For p = 10
−3 = 0.001, plugging in n = 692 gives, to four decimal places, 0.4996, while n = 693 yields 0.5001.
693 is the
lowest common multiple of 7, 9, and 11. Multiplying 693 by 5 gives 3465, the smallest positive integer divisible by 3, 5, 7, 9, and 11.
693 is a palindrome in bases 32, 62, 76, 98, 230, and 692. It is also a palindrome in binary: 1010110101.
The reciprocal of 693 has a period of six: = 0..
693 is a triangular matchstick number.
References
{{reflist
Integers