Understanding Linear Diophantine Equations

The video tutorial explains how to solve a linear Diophantine equation to determine how to make 637 cents using only five-cent and eight-cent stamps. It involves setting up a congruence modulo 5, solving for Y, substituting to find X, and calculating the smallest and largest number of stamps possible. The process is detailed step-by-step, providing a clear understanding of the mathematical concepts involved.

What is the main problem discussed in the video?

• Calculating the total value of stamps needed for a package.

• Finding the number of ways to make 637 cents using 5-cent and 8-cent stamps.

• Finding the number of ways to make 500 cents using 10-cent and 20-cent stamps.

• Determining the cost of sending a letter overseas.

What is the purpose of setting up a congruence modulo 5?

• To determine the exact number of stamps needed.

• To simplify the equation by eliminating the y variable.

• To convert the equation into a quadratic form.

• To find integer solutions for x and y that satisfy the equation.

How do we solve the congruence 3y ≡ 2 (mod 5)?

• By adding multiples of 3 to both sides.

• By multiplying both sides by 5.

• By adding multiples of 5 to 2 until a multiple of 3 is found.

• By subtracting 2 from both sides.

What is the equation for y after solving the congruence?

• y = 4 + 5k

• y = 3 + 5k

• y = 2 + 5k

• y = 1 + 5k

How do we find the smallest number of stamps?

• By using only 8-cent stamps.

• By maximizing the number of 5-cent stamps.

• By using only 5-cent stamps.

• By minimizing the number of 5-cent stamps and maximizing 8-cent stamps.

What is the smallest number of stamps needed to make 637 cents?

• 90 stamps

• 100 stamps

• 80 stamps

• 70 stamps

How do we find the largest number of stamps?

• By using only 5-cent stamps.

• By using an equal number of 5-cent and 8-cent stamps.

• By maximizing the number of 8-cent stamps.

• By minimizing the number of 8-cent stamps.

What is the largest number of stamps that can be used to make 637 cents?

• 115 stamps

• 120 stamps

• 130 stamps

• 125 stamps

What happens if k is set to a negative value?

• The equation becomes unsolvable.

• The number of 5-cent stamps becomes negative.

• The number of 8-cent stamps becomes negative.

• The total value exceeds 637 cents.

What is the relationship between x and k in the solution?

• x = 121 + 5k

• x = 121 - 5k

• x = 121 - 8k

• x = 121 + 8k