Aliquot

[al-i-kwuht]

In mathematics, an aliquot part (or simply aliquot) of an integer is any of its integer proper divisors. For instance, 2 is an aliquot of 12. The sum of all the aliquots of an integer n is the value s(n) = σ(n) - n , where σ(n) is the sum of divisors function.

In chemistry, an aliquot is usually a portion of a total amount of a solution.

The word comes from the Latin aliquoties, "several times".

In pharmaceutics, aliquot refers to a method of measuring ingredients below the sensitivity of a scale. For example, if a scale is inaccurate for samples under 120 mg, but the prescription calls for only 40 mg of drug, an aliquot must be done. This involves adding active ingredient and a proportional amount of diluent to make a "stock" supply. In this case, 120 mg active drug must be weighed and mixed with diluent. Once this stock supply is made, at least 120 mg of this mixture will be taken out and used (as long as this portion contains exactly 40 mg of active drug).

Music

In the construction of string instruments often aliquot parts of the scale length are being used to enhance the timbre of musical instruments. In pianos the aliquot stringing system is sometimes used. Non-Western traditional instruments with sympathetic strings make also use of timbre enhancing based on aliqout stringing and string resonance. The aliquot position (1/7th of the scale length) of the bridge on a violin is also important for the sound of the instrument.

Land Surveying

An aliquot part, in the U.S. Public Land Survey System, is the standard subdivision of area of a section, (a.k.a. a half section, quarter section, or quarter-quarter section). One section of land is a square mile, containing 640 acres (but actual lines as run in the field can produce varying area totals).

Egyptian fraction arithmetic

The aliquot parts of the denominator of the first partition of 2/p conversions to Egyptian fractions was used 25 times in the 1650 BCE RMP 2/n table. The method was in use as late as the Liber Abaci, a text written by Fibonacci in 1202 AD. F. Hultsch first noticed the aliquot part use in 1895. E.M. Bruins confirmed its use in 1945. Today, the use of aliquot parts in Egyptian fraction arithmetic is know as the Hultsch-Bruins method.

Aliquant

By contrast, an aliquant part (or simply aliquant) of a positive integer n is an integer that is not an exact divisor of n. Thus the aliquants of an integer n are the positive integers m less than n such that gcd(m,n) < m. For example, the aliquants of 16 are 3, 5, 6, 7, 9, 10, 11, 12, 13, 14 and 15.

All integers that are less than n but greater than half of n are aliquants of n. Every positive integer less than n is either an aliquant or an aliquot of n, so the number of aliquants of n and the number of aliquots of n sum to n - 1.