256 divided by 7 gives a quotient of 36 and a remainder of 4.

The sum of the digits of 987654 is 39, which is divisible by 9, hence the remainder is 0.

The remainder when a number is divided by 9 can be found by summing its digits. The sum of the digits of 987654321 is 45, which is divisible by 9, hence the remainder is 0.

Using the power of a power property, (a^m)^n = a^(m*n), we have (2^3)^2 = 2^(3*2) = 2^6.

Calculating (3 * 4) = 12, and then 12 mod 5 = 2.

10 divided by 3 gives a remainder of 1, so 10 mod 3 = 1.

Adding 1011 and 1101 in binary results in 11110.

Adding the two polynomials gives (3x^2 + 4x^2) + (2x - x) + (1 + 5) = 7x^2 + x + 6.

Adding the two polynomials gives (3x^2 + 4x^2) + (2x - 3x) + (-1 + 5) = 7x^2 - x + 4.

Adding the coefficients of like terms gives 6x^2 + 2x + 5.

Adding the two polynomials gives (2x^2 + 5x^2) + (3x - x) + (4 + 2) = 7x^2 + 2x + 6.

Adding the polynomials gives (3x^2 + x^2) + (2x - x) + (1 + 4) = 4x^2 + x + 5.

Using the properties of exponents, we combine the exponents: (3x + 2x - 4x) = x, thus the result is 2^x.

Linear programming is used to find the best outcome in a mathematical model whose requirements are represented by linear relationships.

Matrices are used extensively in modern mathematics to represent and solve systems of linear equations, among other applications.

A general partner is actively involved in managing the business and assumes full liability for the partnership's debts.

The managing partner is responsible for overseeing the daily operations and making key decisions for the partnership.

A silent partner provides capital to the partnership but does not take part in the day-to-day management of the business.

A number is divisible by 8 if the last three digits form a number that is divisible by 8.

A number is divisible by 11 if the difference between the sum of the digits in odd positions and the sum of the digits in even positions is divisible by 11.

To check if a number is divisible by 7, you can double the last digit and subtract it from the rest of the number; if the result is divisible by 7, then the original number is as well.

Calculus is significant because it provides essential tools for modeling and understanding change and motion in various fields.

Chaos theory is significant because it explores how small changes in initial conditions can lead to vastly different outcomes in complex systems.

Graph theory is significant as it helps in understanding relationships and networks, which are prevalent in various fields.

Mathematical modeling is significant as it allows for the representation of real-world problems through mathematical expressions, facilitating analysis and solution finding.

The passage highlights that digital sum plays a crucial role in enhancing the accuracy of data interpretation.

The passage highlights that digital sum simplifies the error detection process, making it significant.

Digital sum plays a crucial role in verifying the accuracy of data, ensuring data integrity in various applications.

The passage highlights that understanding historical context is essential for grasping the roots and persistence of inequalities.

The Pythagorean theorem establishes a fundamental relationship between the sides of a right triangle.