9×N

9×n denotes the product of 9 and n, where n is typically an integer. As n varies over the integers, 9×n generates all multiples of 9: …, −27, −18, −9, 0, 9, 18, 27, 36, 45, … If n is restricted to positive integers, the values are the positive multiples: 9, 18, 27, 36, 45, and so on.

In algebraic terms, 9×n is a linear expression in n with slope 9 and passes through the origin. The set of values forms an arithmetic progression with a common difference of 9. Equivalently, the equation 9×n = m has an integer solution n exactly when m is a multiple of 9; then n = m/9.

Divisibility and number properties follow directly from the product form. Every number of the form 9×n is divisible by 9, and every multiple of 9 can be written as 9×n for some integer n. In modular arithmetic, 9×n ≡ 0 (mod 9).

A useful digit-based property is that the sum of the digits of any multiple of 9 is itself a multiple of 9. Consequently, the digit sum of 9×n is a multiple of 9 for all integers n. For example, 9×13 = 117 (digit sum 9), 9×11 = 99 (digit sum 18).

Applications of the concept include studying arithmetic progressions, solving linear Diophantine equations, and patterns in multiplication tables. The expression 9×n serves as a basic example of a simple linear function and of a generator for the set of multiples of 9.