What is an Integer Linear Combination?

An integer linear combination is an expression formed by taking a sum of terms, where each term is the product of an integer and a variable, with all the integers involved being integers. More formally:

Let a and b be integers. An integer linear combination of a and b is any expression of the form ax + by, where x and y are integers.

Explanation:

• Integers: These are whole numbers (positive, negative, or zero). Examples include -3, -2, -1, 0, 1, 2, 3.
• Variables (x, y): These represent integers that can be any integer value.
• The expression ax + by: This represents a sum where each term is the product of an integer (a or b) and a variable (x or y). Because all components are integers, the result of the expression will also be an integer.

Examples:

Let's consider a = 4 and b = 7:

• If x = 1 and y = 1, then ax + by = (4)(1) + (7)(1) = 4 + 7 = 11. 11 is an integer linear combination of 4 and 7.
• If x = -2 and y = 3, then ax + by = (4)(-2) + (7)(3) = -8 + 21 = 13. 13 is an integer linear combination of 4 and 7.
• If x = 0 and y = 5, then ax + by = (4)(0) + (7)(5) = 0 + 35 = 35. 35 is an integer linear combination of 4 and 7.

Generalization:

The concept can be extended to more than two integers. An integer linear combination of integers a₁, a₂, ..., a_n is any expression of the form:

a₁x₁ + a₂x₂ + ... + a_nx_n,

where x₁, x₂, ..., x_n are integers.

In essence, an integer linear combination is a sum of integer multiples of integers.