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In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors.For example, 21 is the product of 3 and 7 (the result of multiplication), and (+) is the product of and (+) (indicating that the two factors should be multiplied together).
An example of a numeral system is the predominantly used Indo-Arabic numeral system (0 to 9), which uses a decimal positional notation. [3] Other numeral systems include the Kaktovik system (often used in the Eskimo-Aleut languages of Alaska , Canada , and Greenland ), and is a vigesimal positional notation system. [ 4 ]
For example, if a = 2 and p = 7, then 2 7 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that a p − 1 − 1 is an integer multiple of p, or in symbols: [1] [2] ().
Cycles of the unit digit of multiples of integers ending in 1, 3, 7 and 9 (upper row), and 2, 4, 6 and 8 (lower row) on a telephone keypad Figure 1 is used for multiples of 1, 3, 7, and 9. Figure 2 is used for the multiples of 2, 4, 6, and 8.
343 = 7 × 7 × 7 = 7 3: the cube of 7, or 7 cubed, wherein replacing two neighboring digits with their digit sums 3 + 4 and 4 + 3 yields 37: 73. Also, the product of neighboring digits 3 × 4 is 12, like 4 × 3, while the sum of its prime factors 7 + 7 + 7 is 21. 307 has a prime index of 63, or thrice 21: 3 × 3 × 7, equivalently 3 × 7 × 3 ...
Here, 3 (the multiplier) and 4 (the multiplicand) are the factors, and 12 is the product. One of the main properties of multiplication is the commutative property, which states in this case that adding 3 copies of 4 gives the same result as adding 4 copies of 3: = + + + =
For example, 3 is a divisor of 21, since 21/7 = 3 (and therefore 7 is also a divisor of 21). If m is a divisor of n , then so is − m . The tables below only list positive divisors.
For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945. Double factorials are used in trigonometric integrals , [ 92 ] in expressions for the gamma function at half-integers and the volumes of hyperspheres , [ 93 ] and in counting binary trees and perfect matchings .