In each of the following pairs is the first number a factors of the second number. 4,72
Answers
In mathematics, a divisor of an integer {\displaystyle n}n, also called a factor of {\displaystyle n}n, is an integer {\displaystyle m}m that may be multiplied by some integer to produce {\displaystyle n}n. In this case, one also says that {\displaystyle n}n is a multiple of {\displaystyle m.}m. An integer {\displaystyle n}n is divisible by another integer {\displaystyle m}m if {\displaystyle m}m is a divisor of {\displaystyle n}n; this implies dividing {\displaystyle n}n by {\displaystyle m}m leaves no remainder.If {\displaystyle m}m and {\displaystyle n}n are nonzero integers, and more generally, nonzero elements of an integral domain, it is said that {\displaystyle m}m divides {\displaystyle n}n, {\displaystyle m}m is a divisor of {\displaystyle n,}{\displaystyle n,} or {\displaystyle n}n is a multiple of {\displaystyle m,}m, and this is written as
{\displaystyle m\mid n,}m\mid n,
if there exists an integer {\displaystyle k}k, or an element {\displaystyle k}k of the integral domain, such that {\displaystyle mk=n}mk=n.[1]
This definition is sometimes extended to include zero.[2] This does not add much to the theory, as 0 does not divide any other number, and every number divides 0. On the other hand, excluding zero from the definition simplifies many statements. Also, in ring theory, an element a is called a "zero divisor" only if it is nonzero and ab = 0 for a nonzero element b. Thus, there are no zero divisors among the integers (and by definition no zero divisors in an integral domain).Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are 1, 2, 4, −1, −2, and −4, but only the positive ones (1, 2, and 4) would usually be mentioned.
1 and −1 divide (are divisors of) every integer. Every integer (and its negation) is a divisor of itself. Integers divisible by 2 are called even, and integers not divisible by 2 are called odd.
1, −1, n and −n are known as the trivial divisors of n. A divisor of n that is not a trivial divisor is known as a non-trivial divisor (or strict divisor[3]). A non-zero integer with at least one non-trivial divisor is known as a composite number, while the units −1 and 1 and prime numbers have no non-trivial divisors.
There are divisibility rules that allow one to recognize certain divisors of a number from the number's digits.7 is a divisor of 42 because {\displaystyle 7\times 6=42}7\times 6=42, so we can say {\displaystyle 7\mid 42}7\mid 42. It can also be said that 42 is divisible by 7, 42 is a multiple of 7, 7 divides 42, or 7 is a factor of 42.
The non-trivial divisors of 6 are 2, −2, 3, −3.
The positive divisors of 42 are 1, 2, 3, 6, 7, 14, 21, 42.
The set of all positive divisors of 60, {\displaystyle A=\{1,2,3,4,5,6,10,12,15,20,30,60\}}A=\{1,2,3,4,5,6,10,12,15,20,30,60\}, partially ordered by divisibility, has the Hasse diagram: